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A monster tale:

a review on Borcherds’ proof of

monstrous moonshine conjecture

Alan Gerardo Reyes Figueroa

Instituto de Matem´atica Pura e Aplicada

IMPA

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A monster tale:

a review on Borcherds’ proof of

monstrous moonshine conjecture

Alan Gerardo Reyes Figueroa

Disserta¸c˜ao

Presented in partial fulﬁllment of

the requirements for the degree of

Mestre em Matem´atica

Instituto de Matem´atica Pura e Aplicada

IMPA

Maio de 2010

A monster tale:

a review on Borcherds’ proof of

monstrous moonshine conjecture

Alan Gerardo Reyes Figueroa

Instituto de Matem´atica Pura e Aplicada

APPROVED: 26.05.2010

Hossein Movasati, Ph.D.

Henrique Bursztyn, Ph.D.

Am´ılcar Pacheco, Ph.D.

Fr´ed´eric Paugam, Ph.D.

Advisor:

Hossein Movasati, Ph.D.

Abstract

The Monster M is the largest of the sporadic simple group s. In 1979 Conway and Norton

published the remarkable paper ‘Monstrous Moonshine’ [38] , proposing a completely unex-

pected relationship between ﬁn i t e simple groups and modular functions, in which related

the Monster to the theory of modular forms. Conway and Norton conjectured in this paper

that there is a close conn e ct io n between the conjugacy classes of the Monster and the action

of certain subgr oups of SL

(R) on the upper half plane H. This conjecture im plies that

extensive information on the representations of the Monster is contained in the classical

picture describing the action of SL

(R) on the u p per half plane. Monstrous Moonshine

is the collection of questions (and few answers) that these observations had dir ect l y inspired.

In 198 8, the book ‘Vertex Operator Algebras and the Monster’ [67] by Frenkel, Lepowsky

and Meurman appeared. This book gave an explanat io n of why the Monster is related to

the the or y of modular forms, by showing th a t it acts as an automorphism group of a vertex

algebra V

♮

of central charge 24, constr ucted in terms of the Leech lattice. Vertex o perators

also arise in mathematical physics contexts, such as conformal ﬁeld theory and strin g the-

ory. Although Fren kel et al. did not prove the Conway-Norton conjecture, th ey provided

the raw material with which this conjecture could be approached. The mai n conjecture was

eventually pr oved by Borcherds in 1992 in his paper ‘Monstrous moonshine and monstrous

Lie superalgebras’ [12]. A key role in th e proof was played by a class of i n ﬁ nite dimen si on a l

Lie algebras —called by Borcherds— generalized Kac-Moody algebras. Borcherds proved

properties of the monster Lie algebra which turned out to be suﬃcient to complete the

Conway-Norton conjecture. A key st ep in the proof was an application of the ‘no-ghost’

theorem from string theory.

Borcherds also obt a ined many other remarkable connections between sporadic ﬁnite simple

groups and modular forms in this and other papers. He was awarded a Fiel d s medal in

1998 for his contributions.

The principal interest of Moonshine is that it cons t it utes a new bridge between al geb r a ic

structures and modular apparatus, and a new era of collaboration between mathematics a n d

physics. This work is a modest revi ew of the Moonshine p henomena: the main conjectures,

Borcherds’ proof, open p r o b l em s and actual areas of research.

vii

Resumo

O Monstro M ´e o maior dos grupos simples espor´adicos. Em 1979 Conway e Nor t on pub-

licaram um remarc´avel artigo ‘Monstrous Moonshine’ [38], onde proporam uma rela¸c˜ao

inesperada entre grupos ﬁnitos simples e fun¸c˜oes modulares, na qual relacionaram o M on -

stro `a teoria de formas modulares. Conway e Nort on conjetura r am neste artigo que existe

uma forte conex˜ao entre as classes de conjuga¸c˜ao do Monstro e a a¸c˜ao de certos subgru-

pos de SL

(R) no semi-plano superi or H. Tal conjetura implica qu e muita informa¸c˜ao

das representa¸c˜oes do M on st r o est´a contida na imagem que descreve a a¸c˜ao de SL

(R)

no semi-plano superior. Monstrous Moonshine ´e a cole¸c˜ao de quest˜oes (e umas poucas

respostas) diretamente inspiradas por essas observa¸c˜oes.

Em 198 8, apareceu o livro ‘Vertex Operator Algebras and the Monster’ [67] de Frenkel, Lep-

owsky and Meurman. Este livro deu u m a explica¸c˜ao de por que o Monstro est´a relacionado

com a teoria de formas modulares, most r an d o que M age como um grupo de automorﬁsmos

de uma ´algebra de vˆertices V

♮

de carga central 24, construida em termos do ret´ıculo de

Leech. Os operadores vˆertice ocorrem tamb´em em contextos da f´ısica matem´atica, tais

como teoria de campos conformes e teoria das cordas. Embora Frenkel et al. n˜ao pr ova r am

a conjetura de Conway-Norton, eles proporcionar a m a ferramenta b´asica com a qual esta

conjetura pode-se provar. A conjetura principal eventualmente foi provada por Borcherds

em 1992 no seu artigo ‘Monstrous moonshine and monstrous Lie superalgebras’ [12]. Um

papel chave na prova ´e feito por uma classe de ´algebras de Lie inﬁnito d i m en si onales

—chamadas por Borcherds— de ´algebras de Kac-Moody g er al iz ad a s. Borcherds provou

propriedades da ´algebra de Lie monstro que foram suﬁcentes para completar a conjetura

Conway-Norton. Um papel fundamental na prova foi uma apli ca¸c˜ao do teorema ‘no-ghost’

da teor i a das cordas.

Borcherds obteve tamb´em muitas outras conex˜oes entre grupos simples espor´adicos e for-

mas modu l a re s em este e outros artigos. Ele foi galardoado com a medalha Fields em 1998

pelas suas contribu¸c˜oes.

O inter´es princi p a l de Moonshine ´e que ele estabele ce uma nova po nte entre as estruturas

alg´ebricas e as modulares, e come¸ca uma nova era de coopera¸c˜ao entre a matem´atica e

a f´ısica. Esta disserta¸c˜ao ´e uma revis˜ao somera do fen´omeno Moonshine: as conjeturas

principais, a p r ova de Borcherds, problemas abertos e ´areas atuais d e pesquisa.

Preface

Monstrous Moonshine is probably one of the most esoteri c achievements ar i sing in m a t h -

ematics. The fact that th e Monster has connections to other parts of mathematics shows

that there is someth i n g very deep going on here. No one fully understands it, and the links

to the ﬁelds of physics and geometry are tantalizing. The Moonshine connections have

spawned a lot of work by a several mathematici ans and mathematical physicists recently,

and h ave opened intriguing conjectures, most of them rem ai ning open until today.

This work grew out in a attempt to explain the mysteries about the Monster and its relati on

with number theory, specially the j modul ar invariant. The idea about work with this topic

was given to me by Prof. Hossein Movasati, with the purpose to have a better understand

on some connections occurring between automorp h i c functions, geometry and physics. The

original i d ea was to write a compl et e and detailed p r oof of the Moonshines conjectu r es, but

by obviously reasons, a complete proof was somewhat voluminous to be given here. I have

decided to restrict attention to the ori gi n a l Conway-Norton conject u r e. Other aspects of

Moonshine are m entioned only as a matter of general cultur e on the subject.

I have written with special attention to the non initiated . In fact, I have includ ed so many

theory on t h e ﬁrst chapters in order to give suﬃcient background to u n derstand al l materi al

in the later ones, and to make easier most of the ideas of Borcherds’ proof. For the sake of

background material, the proofs of theorems and propositions are omitted . The idea is to

give only the proof of the Conway-Norton conjecture, making the material accessible to the

level of a second year graduate student. Unfortunately, this work is by no m ean s complete.

There are several references included, with the purpose of encourage readers to specialize

topics of t heir interest. Perhaps, the recent [72] i s the most complete review of the Moon-

shine phenomenon. A good resume of [72] is the paper [71]. For the non-math em at i ci an

reader, [160] is good sou r ce, basically for its d i vulgation style. Main diﬀerences between

this work and [72, 71] is the devoted attention to the number th e or y of moonshine, and the

detailed exposition of the proof of the Conway-Norton conjecture.

In Chapter 1 I present an historical introduction, in order to facilitate a quickly access to

the main conjectur e. Several concepts appear, most of them explained more detailed in

subsequent chapters.

Chapters 2 and 3 give background material on algebra, particularly on Lie algebras. Chap-

ter 2 also includes other classic material on algebra, such as representation of ﬁnite group s,

modules, tensor products and some con st r u ct i on s . The main topic of Chapter 2 is to ex -

plain aﬃne Li e algebras (Section 2.8), which appear later. Chapter 3 introduce important

concepts in Lie algebra theory, such as the Cartan subalgebras, the root systems and th e

Weyl g r ou p . Main topics of this chapter are the Th eor em of Highest Weights (Section 3.7)

and the Weyl Character formula (Section 3.8).

Chapter 4 introduces the concept of vertex operator algebras, and relate it to th e repre-

sentation theory of Kac-Moody algebras, including a bit of lattice theory. At the ﬁ nal of

Chapter 4, I give some interesting data relating Moonshine to the E

classical Lie algebra

and the Leech lattice.

Chapter 5 is devoted exclusively to a partial construction of the Moonshine module V

♮

given

by Frenkel-Lepowsky-Meurman. It also in cl u des some aspects of other ‘unusual’ algebraic

structures, such as the Golay code C

, the Leech lattice Λ

and the Griess algebra B.

In Ch a p t er s 6 and Chapter 7, I explain in more detail the j-function and the original

Conway-Norton conjecture. This is the only chapter where the proofs are given (mainly

due to the fact that my principal area is number theory, and I have decided give more

attention to this). Chapter 7 includes important useful ma t er i al to understand the ideas of

Borcherds’ p r oof: congruence subgroups of SL

(R), replicable functions and the so called

replication formulae.

Finally, Chapter 8 is devoted to the proof of the Conway-Norton conjecture given by

Borcherds (h er e is where a lot of concepts i ntroduced in previous chapters main t h ei r

contribution). I introduce her e the Borcherds Lie algebras (Sec t io n 8.1) and subsequently

construct the Monster Lie algebra M (Section 8.3), and the denom i n at or formulas (Section

8.4). At the end, the proof is concluded by using properti es of the replication formulae.

Chapter 9 presents the actual status of Moonshine. It includes some generalizations of the

Moonshine conjectures, and explores deep connections between Moonshine, number theory,

geometry and physics. I have add ed a lot o f references to facilitate future work. Most of

the mat er i al of this chapter is t aken from [72].

The order of the chapters is by n o means strict. In fact, most of the material can be

omitted in a rapid lecture. For example, the advanced reader may omit Chapters 2 and 3.

Similarly, r ea d er not interested in number theory can omit Chapter 6. A quickly readin g

of Borcherds proof could be:

Chapter 1; Sections 4.1, 4.3, 4.5, 5.3; Sections 7.3, 7.4, 7.5; a nd Chapter 8

Obviously the ﬁn al chapter is interesting by its own.

Acknowledgments

I would l i ke to express my deep-felt gratitude to Hossein Movasati, for his advice, e n cour-

agement, enduring patience and constant support. He was never ceasing in his beli ef in

me, always pr oviding clear explanations , expert in ind i cat e the simplest way to do th e job,

constantly driving me with energy and always, always, giving me his time, in spite of any-

thing else that was g oi n g on. I wish all students the honor and opportunity to experience

xii

his ab i l ity to perform at that job.

I also wish to thank the other members of my committee, Prof. Henrique Bursztyn from

Impa, Prof. Amilca r Pacheco from UFRJ, and Prof. Fr´ed´eric Paugam from Universit´e

Pierre et Marie Curie - Paris VI. Their suggest i on s , comments and additio n al guidance

were invaluable to the completion of this work . Professor Don Zagier, from Max Plank

Institute of Mathematics, give some useful references about equation (7.16) and made im-

portant comments about future work, so I am i n debt with him. Also, Professor Michel

Waldschmidt from Universit´e Pierre et Marie Curie - Paris VI, made additional useful

comments. I also would tank to Pr ofes sor Alfredo Iusem. He was extrem el y helpful and

provided all the addition al support I needed when things were going not so well.

Additionally, I want to thank all my colleagues: Altemar, Antonio, Arturo, Brun o, Cad´u,

Carlos, Carol, Cristiane, Daniel, Diogo, Guillermo, Jos´e V´ıtor, Juan, L eon a r d o, Lucas,

Marco, Mauricio, Phillip, Rafael, Ricardo, S usana and Tiane. They were always expressing

best wishes and advices. In particular , I appreciate the conversations (and e-mails) with

Arturo, Karla and Mabel, who were always giving their moral support. I must also thank

to Juan, Juan Pablo, Cr i st i an, Mauricio, Pedro and Sergio by th ei r constant advices, an d

to Marlon for his motivation, constant interest and help in the progress of my work. Also,

I must thank Angel, by some suggestions on future work. Thanks to Alvaro for his com-

ments and references in the mathematical physics topics. I would like to thank also Mar´ıa

Jose for her encouragement and her valuable indications about how to use the PsTricks

package for the drawings. Mar´ıa Jose and Carol also g ive some helpful tips in how to

use Beamer. As a special note, I would like to express my gratitude to Al t em a r , who’s pa-

tient and h el p fu l indicat io n s conduced me to complete the proof of the replication formulae.

Finally, I must thank Impa professors and staﬀ for all their hard work and dedication,

providing me the means to complete th i s degree and prepare for a career. I exp r ess my

gratitude to CNPq and Impa for their ﬁnancial support.

A. G. Reyes Figueroa

Rio de Janeiro, maio de 2010.

xiii

Table of Contents

Page

Abstract vii

Resumo ix

Preface xi

List of Tables xix

List of Figures xxi

1 A histo rica l crash course on Monstrous Moonshine 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Finite s im ple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 The discovery of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Modular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Some on Kleinian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 McKay’s obser vation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 The Monst r ou s Moonshine conjecture . . . . . . . . . . . . . . . . . . . . . 14

2 Some bas ics on Lie algebras 17

2.1 Representations of ﬁn i t e groups . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Lie grou ps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Lie algeb r as . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Tensor pr oducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6 Module construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.7 Induced modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.8 Aﬃne Li e algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 The Root space decomposition 49

3.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Complete Reducibility and Semi si m p l e algebras . . . . . . . . . . . . . . . 53

3.3 Cartan subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Roots and Roo t Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 The Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.6 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.7 Integral and Dominant integral elements . . . . . . . . . . . . . . . . . . . 63

3.8 The Weyl character formula . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.9 Representations of SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Vertex algebras and Kac-Moody algebras 81

4.1 Vertex operator algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 The Virasor o algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3 Kac-Moody algebras and t h ei r representation . . . . . . . . . . . . . . . . 87

4.4 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.5 The vertex algebra of an even latt ice . . . . . . . . . . . . . . . . . . . . . 96

4.6 Dynkin Diagrams of classic Lie algebras . . . . . . . . . . . . . . . . . . . . 98

4.7 Lie th eor y and Moonshine . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5 The Moonshine module 105

5.1 The Golay code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2 The Leech lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3 The Monst er vertex algebra V

♮

and the Griess algebra B . . . . . . . . . . 111

5.4 The const r u ct i on of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.5 Improvements o n the Construction . . . . . . . . . . . . . . . . . . . . . . 123

6 Intermezzo: the j function 127

6.1 The modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3 Lattice functions and modular functi ons . . . . . . . . . . . . . . . . . . . 131

6.4 The space of modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.5 The modular invariant j . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.6 Expansions at inﬁ nity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.7 Theta fun c ti o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7 Conway-Norton fundamental conjecture 149

7.1 Hecke operat o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2 Action of T (n) on modul ar functions . . . . . . . . . . . . . . . . . . . . . 152

7.3 Congruence subgroups of SL

(R) . . . . . . . . . . . . . . . . . . . . . . . 153

7.4 Replicable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.5 The repl i cat i on formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8 Borcherds’ proof of Conway-Norton conjecture 165

8.1 Borcherds L i e algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.2 The Borcherds character formula . . . . . . . . . . . . . . . . . . . . . . . 168

8.3 The mons t er Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.3.1 The no-ghost theorem . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.4 Denominator identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

8.5 The twisted denominator identity . . . . . . . . . . . . . . . . . . . . . . . 178

xvi

8.6 Replication formulae again, a n d proof’ s end . . . . . . . . . . . . . . . . . 182

9 Concluding Remarks 185

9.1 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

9.2 Why the Monster? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.3 Other ﬁ n i t e groups: Mini-Moonshine . . . . . . . . . . . . . . . . . . . . . 190

9.4 Modular Moonshine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

9.5 The geomet r y of Moonshine . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.6 Moonshine and physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

References 199

Appendix

A Proof of the Replication Formulae 207

A.1 Faber polynomial s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

A.2 Replicabl e functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

A.3 Deduction of the replication formulae . . . . . . . . . . . . . . . . . . . . . 210

Index 217

xvii

List of Tables

1.1 All 26 ﬁnite sporadic simple groups. . . . . . . . . . . . . . . . . . . . . . . 3

1.2 First d imensions of irreducible characters of M. . . . . . . . . . . . . . . . 13

2.1 Character table for the symmetric group S

. . . . . . . . . . . . . . . . . . 22

2.2 Connectedness properties of so m e classical matrix Lie groups. . . . . . . . 26

2.3 Simply connectedness properties of som e classical matrix Lie groups. . . . 27

2.4 Complexiﬁcations of some classica l real Lie algebras. . . . . . . . . . . . . 33

3.1 Semisimple properties of some classical complex Lie algebras. . . . . . . . . 56

3.2 Semisimple properties of some classical real Lie algebras. . . . . . . . . . . 57

3.3 Roots for the Lie algebra sl

(C). . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4 Roots for sl

, α

}. . . . . . . . . . . . . . 72

3.5 Roots for sl

′

= {α

, α

}. . . . . . . . . . . . . . 73

5.1 Third generation of the Happy Family. . . . . . . . . . . . . . . . . . . . . 118

6.1 First Ber n oulli numbers B

. . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.2 Some val u e s ζ( 2 k) for t he Riemann ζ function. . . . . . . . . . . . . . . . . 141

6.3 Expansion of the ﬁrst Eisenstein series E

. . . . . . . . . . . . . . . . . . . 143

8.1 Graded co m ponents of the Monster Lie algebra M. . . . . . . . . . . . . . 174

8.2 Z-graded components of the Monster Lie algebra M. . . . . . . . . . . . . 175

8.3 Cartan m a tr i x for the monster Lie algebra M. . . . . . . . . . . . . . . . . 177

xix

List of Figures

1.1 The grap h G

555

representing the Bim on st er . . . . . . . . . . . . . . . . . . 7

1.2 Fundamental domain D for the action of P SL

(Z) on H. . . . . . . . . . . 9

3.1 The root system A

for th e Lie algebra sl

(C). . . . . . . . . . . . . . . . . 73

3.2 Roots and domi n a nt integral elements for sl

(C). . . . . . . . . . . . . . . 74

3.3 Roots and domi n a nt integral elements for sl

3.4 The Weyl group for sl

(C). . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 Dynkin diagrams for A

, B

, C

and D

. . . . . . . . . . . . . . . . . . . . 99

4.2 The except i on al Dynkin diagrams E

, E

, F

and G

. . . . . . . . . . . 100

4.3 The aﬃne Dynkin diagrams

and

with l abels. . . . . . . . . . . . 103

6.1 Fundamental domain D and some of its images by S a n d T . . . . . . . . . 128

6.2 Integration on the bound a ry of region D. . . . . . . . . . . . . . . . . . . . 136

6.3 Integration on the bound a ry of region D. . . . . . . . . . . . . . . . . . . . 137

9.1 A ‘friend l y ’ process of compactifying and sewing a 4-punctured plane. . . . 188

xxi

Chapter 1

A historical crash course on

Monstrous Moonshine

1.1 Introduction

In 1978, John McKay made the ob ser vation that

196, 884 = 196, 883 + 1. (1.1)

Here, the number 196,884 refers to the ﬁrst nontrivial coeﬃcient of the automorphic fo r m

or nor m al i zed j-function

J(z) = q

−1

+ 196, 884q + 21, 493, 7 60q

+ . . .

associated to the modular group SL

(Z), that appears in number theory. On the other

hand, the numb er 196,883 refers to the dimension of t h e minimal faithful representation of

the Mon st er group.

The central question of McKay’s equ at i on (1.1) is: What does the j-function (the left

side) have to do with the Monster ﬁnite group (the right side)? In general, specialists on

the subject agree that we still do not und er st and completely this phenomenon. Today we

say that there exists a vertex operator algebra, called the Moonshine module V

♮

, which

interpolates between left and right sides of (1.1): its automorphisms group is the Monster

and its graded dimension is the normalized j-function. Since the original conjecture ap-

peared, the Moonshine quest i on triggered new developments in theory and p u sh e d ou t t h e

boundary of mathematical knowledge, bringing with itself m o r e questions than answers.

This thesis work co n si st s main l y in a modest review of the original proble m and the main

proof of Moonshine conjecture, a brieﬂy look of how this theme is related to other areas of

mathematics, and what are the unsolved questions that remains until today.

The purpose of this chapter is to give a quickly un derstanding of the original Moonshine

conjecture. Because Moonshine is a subject that mixes several branches of mathematics

and physics, the reader is warned that this work is not self-contained an d requires extensive

mathematical baggage. Th er e are several references included, some of them by histor i cal

reasons, others only with th e intention of co n d u c e th e interested r ea d er t o specialize par-

ticular topics.

1.2 Finite simple groups

The discovery of the Monster was preceded by a long history of development of t h e t h eo r y

of ﬁnite groups. It has been of interest to ask for the classiﬁcation of all ﬁni t e groups,

yielding t he enumeration of all kind of symmetries. Remember that a group G is called

simple if it has no non trivial normal subgroup s. Basically, the importance of ﬁnite simple

groups is that by Jordan-H¨older theorem, th ey constitute the ‘primes’ of all ﬁ n i t e groups,

that is, they are the constructive blocks of all ﬁnite grou p s. Thus, the core of the probl em

is the classiﬁcation of ﬁnite simple groups.

This classiﬁcation was announced in 1981 [77]. This result was the culmination of an intense

eﬀort involving several hundred of mathematicians over a period of some 25 years. The

complete classiﬁcation theorem r e qu i r es between 9,000 and 10,000 pages of mathematical

work, although a more streamlined proof is now on progress. By the end of the nineteenth

century, thanks to works of Jordan, Dickson, Chevalley and others, several families of sim-

ple groups were known. In addition, by 1861, Mathieu discovered ﬁve st r an g e ﬁnite simple

groups [141]. This groups were ﬁrst called sporadic in the book of Burnside [23]. Mos t

of the ﬁn i t e simple groups are now called of Lie type or Chevalley groups, and admit a

uniform construct i on in terms of simple Lie algebras, via a systematic treatment discovered

in [27 ].

The modern classiﬁcation race started with the paper of Feit and Thompson in 1963 [61],

showing that every non-cyclic ﬁnite simple group has even order. This gave the great est

impetus to the eﬀort to classify the ﬁnite simple gr o u p s . Thompso n followed this up by

another lengthy pa per in which he classiﬁed al l the minimal simple groups. This papers

made feasibl e the classiﬁcatio n project. In 1972, Gorenstein proposed a strategy for the

classiﬁcation involving a det ai l ed 16 point programme. Progress was rapid from this stage,

with Gorenstein and Aschbacher playing leading roles in the project. It was ﬁnally sh own

in 1981 that every ﬁnite simple group is isomorp hic to one of the foll owing:

• A cyc li c group Z

of pri m e order p.

• An alternating group Alt

for n ≥ 5.

• A si m p l e group of Lie type over a ﬁnite ﬁeld, e. g ., P SL

• Some one of the 26 sporadic simple groups (see Tabl e 1.1).

There are 26 sporad i c simple groups, n ot deﬁnitively organized by a ny simple theme. Fur-

ther d et ai l s about these groups and their classiﬁcation programme can be fou n d in [77].

The largest of the sporadic simple group s is called the Monster. We shall denote this

group by M. The Monster co ntains among its subquotients twenty of the sporadic simple

Group Order Name/Discoverer

·3

·5·11 Mathieu

·3

·5·11 Mathieu

·3

·5·7·11 Mathieu

·3

·5·7·11·23 Mathieu

·3

·5·7·11·23 Mathieu

·3·5·7·11·19 Janko

= HJ 2

·3

·5

·7 Hall-Janko

·3

·5·17·19 Higman-Janko-McKay

HS 2

·3

·5

·7·11 Higman-Sims

McL 2

·3

·5

·7·11 McLaughlin

Suz 2

·3

·5

·7·11·13 Suzuki

He 2

·3

·5

·7

·17 Held

Ru 2

·3

·5

·7·13·29 Rudvalis

·3

·5

·7

·11·13·23 Conway

·3

·5

·7·11·23 Conway

·3

·5

·7·11·23 Conway

·3

·5

·7·11·23 Fisch er

·3

·5

·7·11·13·17·23 Fischer

Fi’

·3

·5

·7

·11·13·17·23·29 Fischer

O’N 2

·3

·5·7

·11·19·31 O’Nahn

Ly 2

·3

·5

·7·11·31·37·67 Lyons

·3

·5·7·11

·23·29·31·37·43 Janko

HN 2

·3

·5

·7·11·19 Harada-Norton

Th 2

·3

·5

·7

·13·19·31 Thompson

B 2

·3

·5

·7

·11·13·17·19·23·31·47 Baby Monst e r

M 2

·3

·5

·7

·11

·13

·17·19·23·29·31·41·47·59·71 The Mons t er

Table 1.1: All 26 ﬁnite sporadic simple groups.

groups, except for J

, Ru, O’N, Ly and J

. These twenty constitute the Happy Family,

and they occur naturally in three generations: the family of Mathieu groups, the family

of Conway groups, and the so called Third Generation. Each one of these is related to

an algebraic structure, in particular, the last one is associated to the Monster. The other

ﬁve sporadic groups a re called the pariah. Fischer and Griess i n dependently predicted in

1973 the existence and properties of M as t he lar g est of the sporadic groups. It has 194

conjugacy classes and irreducible characters. The character table of M was determined

by F ischer, Livingstone and Thorne in 1978 [62] and can be found in the Atlas of Finite

Groups [ 36]. The degree of the smallest i r r ed u ci ble characters of M are:

= 1, d

= 196883, d

= 2129687 6, d

= 8426093 26, . . .

In part i cu l a r, the minimal faithful representation of the Monster would have dimension

= 196, 883. (1.2)

A lot of information about M can be fou nd in [36].

As a remarkable fact, we mention that the greatest of the Mathieu sporadic groups, M

was

constructed by Mathieu as the grou p of symm et r i es of th e Golay code C

(a 12-dimensional

subspace of a 24-dimensional vector space over F

). Similarly, Conway constructed his 3

sporadic groups in 1968, and the greater of them, Co

, is realized as the automorphism

group of the Leech lattice Λ

(a 24- d i m en s io n al subspace of the 26-dimensional Lorentzian

vector space R

25,1

, wi t h norm ||x||

= x

+. . .+x

−x

). Norton observed that the minimal

representation of M would have the structure of a rea l commutative non-associative algebra

with an associative form . Finally, the Monster group was ﬁrst constructed by Griess in

1980 as a group of automorphisms of a commutative non-associative algebra of dim en si on

196,883 over Q with an associative form [80]. Griess’ construction has been simpliﬁed i n

works of Tits [174, 175, 176] and Conway [34, 35]; and Tits has in fact proved that M is

the full group of automorphisms of the Griess algebra B. Unfortunately, even this ﬁne

version of Griess algebra does not appear as elegant as Golay code C

or Leech lattice Λ

which have simple characterizations. But, there was some hints that the Monster could be

associated with an elegant canonical str u ct ure. We shall discuss this structure in Chapter

1.3 The discovery of M

We have already mention that Feit and Thompson’s p aper [61] was the starti n g point on

the syst em at i c search of sporadic groups, or sometimes called symmetry atoms, leading

to a whole project known as ‘t h e Classiﬁca ti o n ’ [77]. The idea: to compil ing a list of all

ﬁnite symmetry atoms, and showing that the l i st was complete (like a periodic table of

symmetries). Recall that there were already know some of this exceptional the sporadic

simple groups. Conway constructed his three sporadic groups Co

, Co

and Co

in 1968,

by loo ki n g various mirror symmetries occurri n g within Leech lattice Λ

, and this lattice

had yielded a tot a l of 12 sporadic groups, nine of which had already appeared elsewhere.

These 12 groups, along with Janko’s groups J

and J

—which had nothing to do with the

Leech lattice— brought the total number of exceptions to 14. By the end of 1972 there

were six more: thr ee found by Fischer; one by Dieter Held; one by Richard Lyons; and one

by Ar u n as Rudvalis. Both Held and Lyons used the cross-secti o n meth od. Rudvalis used

permutations. Th e tot al number o f excep t i on s was now 20. With so much activity and so

much information coming in, it made excellent sense to collect it al l, correct errors, and

present it in a form that was easy to read and readily available. This was ‘the Atlas’ [36]

(exceptions to periodic table).

In fact, Fischer constructed h i s sporadic groups Fi

, Fi

and Fi’

by using the method of

transpositions (adding some new symmetries to the Mathieu groups M

, M

respec-

tively), leading him to the discovery of the Baby Monster B in 1973. Another new group

of symmetries was found by Michael O’Nan the same year. This brought the total to 22,

though not all these groups were yet known to exist. After that, it was expected that no

other new sporadics would appear. Fortunately, Fischer and Griess noticed independently

that this Fischer’s huge new group could appear as a cross-sectio n in a larger group. If a

group emerged from the cross-section method —like B— then a great deal of information

needed to be calculated before a construction was possible. Most of this data was encoded

in the form of a square array of numbers called a character table (a character table is a

square array of number s that that express the fund am entally ways the group can operate in

multidimensional space) (see Section 2.1). In fact, a ﬁnite group can have a huge character

table, but sporadic groups usual ly have a low number o f rows/columns in their character

tables.

In late 1973, knowing only that the Monster had two cross-section s, using a procedure

called Thompson order formula, the size of the whole thing was within reach. Thompson’s

technique needed detailed computations on how the two cross-sections could intersect.

Fischer used it to show that the size could not be greater than a certain number. Further

calculations made by Conway, led Fischer to four n ew sporadic groups: B, Th, HN, and

M, the last one with size

· 3

· 5

· 7

· 11

· 13

· 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71

Having the size of the Monster was essential before working out it s character table. Nor-

ton and others established that the Monster would have a representation of dimen si on

196, 883 = 47 · 59 · 71. Only with this information, Fischer, Livigstone and Thorne com-

puted th e entire character table of M, in 1974; and then that of the Baby Monster. Final l y,

Sims and Leon co n cl u d e the construct i o n of B in 1977, by creating it on a computer as a

group of permutations on 13,571,955,000 mirrors.

After Sims and Leon constr u ct i on , it was natural to ask whether the Monster could be con-

structed in a similar way. Un fo rt unately this seemed to be out of sight, so an alternative

method was needed. Perhaps one could use multidimensional space. Similar methods had

been applied to other exceptional groups, such as Janko’s ﬁrst group J

. But, as J

needed

seven dim en s io n s, th e Monster needed 196,883, which means that a single operation in the

Monster would appear as a matrix with nearly 196,883 rows and as many columns.

Suddenly in 1980, Bob Griess announced a cons t ruction. Alread y, Nort on had ﬁgure out

that the Monster must preserve an algebra structure in 196,884 dimensions. This structure

would allow any two points to be multiplied together to give a thir d point. Griess’ ﬁrst

task was to construct a suitable multiplication, solving some problems about signs on th e

algebra structure, by tracking them back into the grou p . This group Gr ie ss i s r efer r i n g to

here is a huge subgroup of the Monster, ext ended by a factor of over 32 million. This group

needs 96,308 dimensions, and is one of the two cross-sections of the Monster (the other

one involves the Baby Monster). Fischer used it earli er in helping to build the character

table of the Monster, and Griess now us ed it to help to const r uct the Monster in 196,884

dimensions. He knew that the action of this huge subgrou p must split the space into three

subspaces of the following dimensions (see (5.19) and (5.20)):

98, 304 + 300 + 98, 280 = 196, 884

The ﬁrst number is 98, 304 = 2

× 24. This is the space needed for the cross-section

mentioned above. The second number is 300 = 24 + 23 + 22 + ... + 3 + 2 + 1. This c om es

from a triangular arrangement of numbers with 24 in the ﬁrst row, 23 in the second row, 22

in the th ir d row, and so on. The third number i s 98, 280 =

(196, 560). This comes from

the Leech lat t i ce (see Chapt er 5), where there are 196,560 points closest to a given point,

and they come in 98,280 diametrically opposite pairs. Each pair yields an axis through the

given point, and in the 196,884-dimensional space these axes become i n d ependent of one

another. Griess [80] ﬁnally sort out the sign problem for the multiplicat i on , an d proved

that the group of symmetries contains the Monster, by addi n g on e ext r a permutation to

create a larger group.

Two other mathematicians got deeply involved in looking at the Gr i ess construction. One

was Jacques Tits, who found a way of avoidi n g the sign problems [ 17 5 ] . Tits also found a

number of other impr ovements and simp l i ﬁ ed Griess’s construction. The other person who

took a detailed interest in the Griess construction was Conway, giving a construction of his

own, and a very elegant proof of ﬁniteness [34, 35]. In a sense, Tits avoided all calculations

in the Monster, while Conway made easier to per for m such calculations. Conway’s con-

struction is similar to Griess’s in the sense that they both use the same large cross -s ect io n of

the Monster to get started. Thi s splits the 196,884-dimensional sp ace i nto three subspaces.

A more detailed (with a lot of hist o r ica l notes) on the construction of the sporadic grou p s

can be found in [160]. The interested reader may also see [72] to have a complete context

on the Moonshin e phenomenon. The compl et e construction of the Monster group is too

voluminous to be explained here . We will present a sketched part i al construction of M in

Section 5.4.

For now, we describe a remarkably simple representation of M. As with any noncy cli c ﬁnite

simple group, it is generated by its involutions (i. e., elements of order 2) and so will be

an image of a Coxeter group. Let G

pqr

, p ≥ q ≥ r ≥ 2, be the grap h consisting of three

strands of lengths p + 1, q + 1, r + 1, sharing a common endpoint. Label the p + q + r + 1

nodes as in Figure 1.1. Given any graph G

pqr

, d eﬁne Y

pqr

to be the group consisting of a

generator for each node, obeying the usu al Coxeter group relations (i. e., all generators are

involutions, and the product gg

′

of two genera t or s has order 2 or 3, depending on whether

or not the two nodes are adjacent), together wi t h one more relation:

(ab

)

= 1.

The groups Y

pqr

, for p ≤ 5, ar e all i d entiﬁed (see [102]). Conway conjectured and , building

on work by Ivanov [103], Norton proved [156] that Y

555

∼

444

is the Bimonster, the

wreathed-square M ≀Z

(or (M ×M).2 in Conway’s notation), that is, a group with M ×M

as normal sub g ro u p and Z

as quotient, with order 2|M|

. A closely r e la t ed presentation

of the Bimonster has 26 involuti on s as genera t or s and h as relations given by the incidence

graph of the projective plane of order 3; the Monster itself arises from 21 involutions and

the aﬃne plane of order 3. Likewise, Y

553

∼

443

∼

M × Z

. Other sporadic groups arise

in e. g., Y

533

∼

433

∼

B is the Baby Monster; Y

552

∼

442

is the Fischer grou p Fi’

; and

532

∼

432

is the Fischer group Fi

. The Coxeter group s associated to the other g r ap hs

pqr

, p ≤ 5 are all ﬁnite groups of hyperbolic reﬂections in R

17,1

and contain copies of the

Weyl group E

, giving a rich underlying geometry.

Figure 1.1: The graph G

555

representing the Bim o n st er .

1.4 Mod ul ar fu ncti on s

Although the Monster group M was discovered within the context of ﬁnite simpl e grou ps,

hints later began t o emerge that it might be strongly related to other bran ches of mathe-

matics. One of these is the theory of mod u l ar function s and modular forms.

Consider the action of the group SL

(R) on the upper half-plane H. Let

(R) =



a b

c d



∈ GL

(R) : ad − bc = 1



(R) acts on H = {z ∈ C : Im(z) > 0} by



a b

c d



· z =

az + b

cz + d

(note that Im(z) > 0 ⇒ Im



az+b

cz+d



= Im(z) > 0). In particular, the subgroup SL

(Z) acts

on H discontinu a ll y.

Since



−1 0

0 −1



· z =

−z

−1

= z, we see that P SL

(Z) = SL

(Z)/{±I} acts on H. We call

Γ = P SL

(Z) the modular group and we denote by H/SL

(Z) the set of orbits. Since

T =



1 1

0 1



∈ SL

(Z) an d



1 1

0 1



· z = z + 1,

then z and z + 1 are in the same orbit. Thus, each orbi t intersects

{z ∈ H : −1/2 ≤ Re(z) ≤ 1/2}.

Similarly,

S =



0 1

−1 0



∈ SL

(Z) an d



0 1

−1 0



· σ = −

so the elements σ and −1/σ ar e in the same orbit. Thus, each orbit intersects

{σ ∈ H : |σ| ≥ 1}.

In particular, if |σ| = 1, then (

0 1

−1 0

) ·σ = −

= −¯σ. In fact, Theorem 6.1.2 shows that we

obtain a fundam ental domain D for the action of SL

(Z) on H taking the region

D = {z ∈ H : −1/2 ≤ Re(z) ≤ 1/2, |z| ≥ 1},

and identifying z with z + 1, for Re(z) = −1/2; and σ with −¯σ = −

, for |σ| = 1 (see

Figure 1. 2 ) .

−1

−1/2

1/2

τ + 1

−1/σ

Figure 1.2: Fund am ental domain D for the action of P SL

(Z) on H.

With these identiﬁcations we obtain a set D intersecting each orbit just at one poi nt. In

fact, the canonical map D → H/SL

(Z) is surjective and its restricti on to the interior of

D is inject i ve. We can prove also that the modular group Γ is generated by S and T (see

Theorem 6.1.3). The set H/SL

(Z) of orbi t s has the structure of a Riemann su r fa ce with

one point removed. This is a Riemann su r face of genus 0. When we rem ove one point of it,

we obtain a set which can be identiﬁed with C. Thus, we have an isomor p h i sm of Riemann

surfaces

H/SL

(Z) → C.

This ma p can be extended to an iso m or p hism between compact Riemann surfaces

H/SL

(Z) ∪ {i∞} → C = C ∪ {∞}

∼

which maps i∞ → ∞. Such an i som o r p h i sm is not unique. However, when we operate

with Riemann surfaces of genus 0, we can prove that if j is just one of these isomorphisms,

then any other is of the form a(j + b), where a, b are constants and a = 0.

Let j : H/SL

(Z) →

C be one of such isomorphisms. Then, j deﬁnes a map j : H → C that

is constant on orbits. Since z and z +1 l ies on the same orbit, then we have j(z) = j(z + 1).

Thus, j is periodic (of period 1). This implies that j has a Fourier expansion

j(z) =



n∈Z

2πinz

Writing q = e

2πiz

, then we have

j(z) =



n∈Z

Let k be an integer. We say tha t a mer om o r p h i c function f : H → C is a modular function

of weight 2k i f



az + b

cz + d



= (cz + d)

f(z), for all



a b

c d



∈ SL

(Z).

If f is holomorp h i c everywhere —including at inﬁnity—, we say t hat f is a modular form.

We give some examples (see Section 6. 3 ) :

Example 1.4.1. Let σ

(n) =



d|n

. We deﬁne the Eisenstein series

(z) = 1 + 240



n≥1

(n)q

= 1 + 240q + 2160q

+ . . .

Thus, E

(z) is a modular form of weight 4.

Example 1.4.2. The Dedekind’s function

η(z) = q



d≥1

(1 − q

)

= q − 24q

+ 252q

− . . .

is a modular form of weight 12.

We now deﬁne j : H → C by

j(z) =

(z))

η(z)

This is a modular form of weight 0, so it is constant o n orbits of P SL

(Z) on H. In fact,

j(z) = q

−1

+ 744 + 196, 884q + 21, 493, 760q

+ . . . (1.3)

Such map s are usually called fundamental functions or Hauptmodul.

Observe that j(z) i s holomorphic on H and t h a t j(z) has a simple pole at z = i∞ (i . e., at

q = 0). Then, j gives an holomorphic isomorphism between H/SL

(Z) and C that can be

extended to a meromorphic isomorphism of compact Rieman n surfaces

H/SL

(Z) → C,

where

H = H ∪ Q ∪ {i∞}.

The SL

(Z)-orbits of Q ∪ {i∞} are called cusps and thei r role is to ﬁll the punctures

Some authors say that f is of weight k.

of H/SL

(Z), compactifying the surface, as there ar e much fewer meromorphic functions

on compact surfaces than on noncompact ones. Since any other isomorphism is of the

form a(j(z) + b), with a, b constants and a = 0, in particular there is just one of such

isomorphisms with leading coeﬃcient c

= 1 and constant term c

= 0:

J(z) = j(z) − 744 = q

−1

+ 196, 884q + 21, 493, 760q

+ . . . (1.4)

We shall call this function J(z) the canonical isomorphism or the normalized Hauptmodul

of the modular group.

The expansion coeﬃcients of J(z), which are all positive integers (except for the constant

term), mig ht appear unattractive. As we shall see, i t took many years and an accident

before their mea n i n g was ﬁnally found.

1.5 Some on Kleinian groups

Even before the discovery of the modular invariant j(z) was made, it was observed th a t

certain characteristics of ell i p t i c functions with per i ods ω

, ω

were invariant only under a

certain sub gr o u p Γ(2) of Γ = SL

(Z). This and other facts led Klein to th e creation of

the theory of congruence subgroups [115]. He introduced a class of principal congruence

subgroups

Γ(n) =



a b

c d



∈ Γ :



a b

c d



≡



1 0

0 1



(mod n)



, for n > 0,

and notions of congruence subgroups Γ

′

of level n. An example of a congruence subgroup

of level n is the cla ss

(n) =



a b

c d



∈ Γ : c ≡ 0 (mod n)



At the same time, Poincar´e, inﬂuenced by a paper of Fuchs [69], launched a p r ogr a m to

study discrete subgroup of P SL

(R) and their corresponding automorphic functions (anal-

ogous t o modul ar functions) , and this laid to the theory of Fuchsian groups.

One of the basics results of this theory is th at for any Kleinian group Γ

′

, a suitable com-

pactiﬁcation of H/Γ

′

has the structure of a compact Riemann surface. In the special case

when the genus of H/Γ

′

is zero the theory of automorphic functions becomes specially

simple: the ﬁeld of automorphic functions is generated by only one funct i on J

′

(z), called

the Hauptmodul of Γ

′

. In the par t i cu l ar case of the modular group Γ = P SL

(Z), the sur-

face —the Riemann sph e r e— has genus zero, and the Hauptmodul of Γ is just J(z) in (1.4).

Fricke [68] investigated the surfaces associat ed with Γ

(n). In particular, congruence sub-

groups Γ

(p), for p prime, provide examples of genus zero surfa ces if and only if p −1 | 24.

One can obtain more examples adjoining to Γ

(n) the Fricke involuti on w

(z) = −1/nz,

which of course can be realized as an element of P SL

(R). Th a t is

(n)



(n),

√



0 −1

n 0



The normalizer of Γ

(n) in P SL

(R) was fully described by Atkin and Lehner [3]. When

n is a prime p, it is just the group Γ

(p)

generated by Γ

(p) and the Fricke involut i on

. Ogg [158] completed Fricke’s proof [68] that for a prime p, Γ

(p)

has the genus zero

property if and only if

p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 . (1.5)

This strange set of primes could be passed to histor y as another mathematical fact without

any special sign i ﬁ c an ce. It hap pened however that Ogg attended a talk of Tits mentioning a

certain sporadic sim p l e group predicted —bu t not proved— to exi st by Fischer and Griess,

of order

|M| = 2

· 3

· 5

· 7

· 11

· 13

· 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71,

and noticed that the primes app ea r i n g in l i st (1.5) are exactly the prime divisors of the

order of the Mon st er group M. What a coincidence! It was not realized at th at time, at

1975, t h a t this coincidence was the tip of an iceberg.

1.6 McKay’s observation

Hints that the Mon s te r might in fact be associated with an elegant structure had appeared

before Griess announced h is construction. We had mention already that Ogg, who was

working on the ﬁeld of modular fun ct i ons, came with some coincidences. From the other

side, McKay, who was working in ﬁnite group theory, noticed an o t h er relation between t h e

Monster and modular functions: the near coincidence of the minimal possible representation

of M (1.2) and the ﬁrst non triv i al coeﬃcient c

of the Hauptmodul J(z) in (1.4).

196, 884 = 196, 883 + 1.

Soon, McKay and Thompson found similar relations [172] includ ing anot h er di m en si ons of

irreducible representations of M, for example:

196, 884 = 196, 883 + 1,

21, 493, 760 = 21, 296, 876 + 196, 883 + 1,

864, 299, 970 = 842, 609, 326 + 21, 296, 876 + 2 · 196, 883 + 2 · 1, (1.6)

If we interpret d

= 1 as the dimension of the trivi al representation of M, then we have

= d

+ d

where d

is the dimension of the next irreducible representation of the Monster. In fact, if

we comput e some more few coeﬃcients of J(z)

J(z) = q

−1

+ 196, 884q + 21, 493, 760q

+ 864, 299, 970q

+ 20, 245, 856, 256q

+ 333, 202, 640, 600q

+ 4, 252, 023, 300, 096q

+ . . .

and som e more few dimensions of irreducible characters of M

= 1 d

= 196, 883

= 21, 296, 876 d

= 842, 609, 326

= 18, 538, 750, 076 d

= 19, 360, 062, 527

= 293, 553, 734, 29 8 . . .

Table 1.2: First dimensions of irreducible characters of M.

then eq u at i ons (1.6) are extended:

= d

+ d

= d

+ d

= 2d

+ 2d

+ d

= 2d

+ 3d

+ 2d

+ d

= 4d

+ 5d

+ 3d

+ 2d

+ d

, (1.7)

. . .

and further relations of this sort. Based on these observat io n s, McKay and Thompso n

conjectured the exist en ce of a natural inﬁnite-dimensional representation of the Monster

V = V

−1

⊕ V

⊕ . . . ,

such that dim V

= c

, for n = −1 , 1, 2, 3, . . .. That is, this suggests that there is a graded

vector space acted on by the Monster, and that our Hauptmodul J(z) is in fact what we

call t h e graded dimension of V :

J(z) =



n≥−1

= q

−1



n≥1

(dim V

. (1.8)

McKay, also gave a r el at i o n between the j-function and the classical Lie algebra E

(see

Section 4.7) . A more elementary observation concerns the Leech lattice Λ

. The Leech

lattice is a particularly special one in 24 dimensions. It has some special features. The

book by Conway and Sloane [39] includes a lot o f interesting details relating the Leech

lattice. Another useful reference is [86] . For example, 196,560 is the number of vectors in

the Leech lat ti ce with (squa r ed ) norm equal to 4, an d note that this number is also close

to 196,884: In fact, the ﬁrst coeﬃcients of the Hauptmodul J (1.4) are related to the Leech

lattice

196, 884 = 196, 560 + 324 · 1,

21, 493, 760 = 16, 773, 120 + 24 · 19 6, 560 + 3, 200 · 1,

864, 299, 970 = 398, 034, 000 + 24 · 16, 773, 120 + 324 · 196, 560 + 25, 650 · 1,

where 16,773,120 and 398,034,000 are the numbers of 6-norm and 8-n or m vectors in Λ

This may not seem as convincing as (1.6), but the same equations hold for any of the

24-dimensional even self-dual lattices, apart from an extra term on the right sides corre-

sponding to 2-norm vectors (there are none of th e se in the Leech).

What conceptually does the Monster, the Leech lat t ice have to do with the j-function?

Is there a common theory explaining this numerology? The answer is yes. In fact, as we

shall see, there is a relation between E

to the j-function. In the late 1960’s, Victor Kac

[110] and Robert Moody [1 49 ] independently deﬁned a new class of inﬁnite-dimensional

Lie algebras. A Lie algebra is a vector space with a bilin ear vector-valued product that is

both anti-commutative and anti-associative (Chapter 2). The familiar vector-product u ·v

in three dimensions d eﬁnes a Li e algebra, called sl

(R), and in fact this algebra generates

all Ka c-M oody algebras. Within a decade it was realised that the graded dim en si o n s of

representations of the aﬃne Kac-Moody algebras are (vector-valued) modular functions for

(Z) (se e Theorem 3.2.3 in [72]).

1.7 The Monstrous Moonshine conjecture

Recall the strong connections of previous section, that suggested the existence of the graded

algebra V . Thompson [171] also proposed considering, for any element g ∈ M, the modular

properties of t h e series

(z) =



n∈Z

tr(g|V

= q

−1

+ tr(g|V

)q + tr(g|V

+ . . . , (1.9)

where q = e

2πiz

and V

is t h e n-th graded component of V . This graded trace above is

called the McKay-Thompson series of g, and generalizes our Haup t m odul on (1.4). For

example, if g = 1 (the identity element of M), then T

(z) is just J(z).

Working with data fr om the table character of the Monster, remarkable numerology con-

cerning these graded t r aces was collected in the paper Monstrous Moonshine [38 ]. It has

been pointed by Conway that he found some of these interesting series in the classical book

of Jacobi [106].

Inﬂuenced by Ogg ’s observat i on , Thompson, Conway and Norton realized that all the se-

ries they were discovering (proceedin g exper i m entally by the ﬁrst few coeﬃcients) were

normalized gener at o rs of genus zero function ﬁelds arising from certain discrete subgroups

of P SL

(R), th at is the McKay-Thompson series they were discoverin g beh ave as ‘mini

j-functions’, for certain other subgroups of P SL

(R). They were led to conjecture that

there exists a graded representation V of t h e Monster —in fact a double- gr aded space—

with all the functions T

(z) having this genus zero property. Knowing the function s T

(z)

determines the M-m odule V uniquely, and the qu est i on was whether it existed, given the

list of proposed functions T

(z), for each of the 194 conjugacy classes of M. Such graded

module was subsequently discovered by Frenkel, Lepowsky and Meurman [66], [67]. It is

called the monster vertex algebra or the moonshine module V

♮

. We shall study it in Chap-

ter 5.

Instead of consider only the modular subgroup P SL

(Z) of PSL

(R), we shall consid er

other subgroups. A subgroup G of P SL

(R) is called commensurable with P SL

(Z) if:

• [P SL

(Z) : P SL

(Z) ∩ G] is ﬁnite;

• [G : P SL

(Z) ∩ G] is ﬁnite.

We may consider the action of such a group on the up per half-plane H. We know that if G

is a subgroup of P SL

(R) commensu r able with P SL

(Z), then the set of orbits H/G has

the structure of a compact Riemann surfa ce, with ﬁnitely many points removed. We write

H/G ⊆

H/G,

where

H/G is a compact Riemann surface. Let H/G a compact Riemann surface of genus g.

Then,

H/G is homeomorphic to a g-torus. In the case g = 0, H/G is homeomorphic to the

Riemann sph er e CP

. A subgroup G is called of genus 0, if the Riemann sur face

H/G has

genus 0. We have already pointed in Section 1.5 that under these particular circumstances,

the ﬁeld of automorphic functions of

H/G → CP

is generated by just one element, and

we can take this element as the unique isomorphism of Riemann surfaces

H/G → CP

with leading coeﬃcient 1 and con s t ant term 0. We denote this by J

(z), and is called the

canonical isomorphism or normalized Hauptmodul of G. So J

, plays exactly the same role

for G that the J-function plays for SL

(Z). For example, i n t h e case of Γ

(2), Γ

(13) and

(25) —ar e all genus 0 subgroups of P SL

(R)—, we obtain Hauptmoduls

(2)

(z) = q

−1

+ 276q − 2048q

+ 11202q

− 49152q

+ 184024q

+ . . . , (1.10)

(13)

(z) = q

−1

− q + 2q

+ q

+ 2q

− 2q

+ q

+ . . . , (1.11)

(25)

(z) = q

−1

− q + q

+ q

− q

+ q

− q

+ . . . (1.12)

We now state formally an in i t ia l form of the Conway-Norton conjecture, or Moonshine

conjecture.

Conjecture 1.7.1 ( Moonshine Conjecture). (Conway-Norton, 1979).

For each g ∈ M, the McKay-Thompson series T

(z) is the normalized Hauptmodul J

(z) :

H/G → CP

for some subgroup G of SL

(R) commensurable with P SL

(Z).

The ﬁrst major step in the proof of Monstrous Moonshine was accomplished in the mid

1980’s with the construction by Frenkel-Lepowsky-Meurman [67 ] of th e Moo n sh i n e module

♮

, and its interpretation by Richard Borcherds [8] as a vertex operator algebra ( VOA). In

1992, Borcherds [12] com p l et ed the proof of the original Monstrous Moonshine conjectures

by showing that the graded characters T

of V

♮

are indeed the Hauptmodu l s identiﬁed by

Conway and Norton, and hence that V

♮

is indeed the desired representation V

♮

of M conjec-

tured by McKay and Thompson. The algebraic structure appearing in moonshine typically

arises as the symmetry group of the associated vertex operator algebra, for exam p l e, that

of V

♮

is the Monster M. By Zhu’s Theorem ( Th eor e m 9.2.1), the modular functions appear

as graded dimensions of the (possibly twisted) modules of a vertex operator alg eb r a. In

particular, the answer that this framework provides for what M, E

and the Leech latt ic e

have to do with the j-function i s that they each correspond to a vertex operator algebra

with a single simple module; their relation to j is then an immediate corollary to the much

more gen e ra l Zhu’s Theorem.

Moonshine is also profoundly connected with geometry a n d physics (namely conformal ﬁeld

theory and string theory). Stri n g theory proposes that the el em entary part i cle s (electr ons,

photons, quarks, etc.) are vibrational mod es on a strin g of length about 10

−33

cm. These

strings can interact only by splitting apart o r joining together as th e y evolve through time,

these (classical) strings will tr a ce out a surface called the world-sheet. Quantum ﬁeld theo r y

tells us that the quant u m quantities of interest (amplitudes) can be perturb a t ively com-

puted as weighted averages taken over spaces of these world-sheets. Conformally equivalent

world-sheets should be identiﬁed, so we are led t o i nterpret amplitudes as certain integrals

over moduli spaces of surfaces. This approach to stri n g theory leads to a conformally in-

variant quantum ﬁeld theory on two-dimensional space-time, called conformal ﬁeld theory

(CFT). The various modular forms and functions arising in Moonshine appear as inte-

grands in some of these genus 1 surfaces appearing i n these conformal theories. We shall

explain a bit of these relation between Moonshine and physics in Ch apter 9. Thus, the

actual importance of the Moonshine pheno m en on: It proposes a conexion between four

branches of math em at i cs, namely algebraic structures, modular structures, geometry and

mathematical physics.

Chapter 2

Some basics on Lie algebras

In this chapter we introdu ce Lie algebras termed aﬃne, of which Virasoro algebra plays

a central role throughou t this work. We also present standard constructions of important

classes of modules and algebras and we discuss the concept of graded dimension. We

have introduced a number of elementary concep ts i n ﬁrst sections to provide suﬃcient

background for understanding the main concepts. For an extensive exposit i on of b asi c

algebra, the reader may refer to [107, 108] and [127]. A detailed reference on Lie groups

and Li e algebras can be found in [100] an d [179], or the more recent [91 ] .

2.1 Representations of ﬁnite groups

Let V be a vector space over the ﬁeld C of complex numbers and let GL(V ) be the group

of isomorphisms of V onto itself. An element a ∈ GL(V ) is, by deﬁnition, a linear mapping

of V into V which has an inverse a

−1

; this inverse is linear . When V has a ﬁnite basis (of

n elements), each linear map a : V → V is deﬁned by a square matrix [a

] of order n. Th e

coeﬃcients a

are complex numbers, and we usually write in this case GL(V ) as GL

(C).

Suppose now G is a ﬁnite group, with i d e ntity element 1 and with composition (s, t) → st.

A linear representation of G in V is a homomorph i sm π from the group G into the grou p

GL(V ). In other words, we associate with each element s ∈ G an element π(s) of GL(V )

in such a way that we have the equality

π(st) = π(s) · π(t), for s, t ∈ G.

Observe that the preceding formula implies the following: π(1) = 1, and π(s

−1

) = π(s)

−1

When π is given, we say that V i s a representation space of G (or even simply, a repre-

sentation of G). We restrict ourselves to the case where V has ﬁnite dimension. When

dim V = n, we say that a representation π : G → GL(V ) has degree n.

Let π : G → GL(V ) be a linear representation and let W be a vector subspace of V . We

say that W is stable under the action of G (we say also invariant), if π(s)W ⊆ W , for

all s ∈ G. Given a linear representation π : G → GL(V ), we say that it is irreducib le or

simple if V is not 0 and if V has no nontrivial stable subspa ces und er G. This condition is

equivalent to saying V is not the direct sum of two representations (except for the trivial

V = 0 ⊕ V ). We have the following result [166, p . 7]

Theorem 2.1.1. Every representation is a direct sum of ir reducible representations.

Let V be a vector space having a basis e

, . . . , e

, and let a be a linear map of V into itself,

with m at r i x [a

]. By the trace of a we mean the sca la r

tr a =



1≤i≤n

It is the sum of the eigenvalues of a (counted with their multiplicities), and does not depend

on the choice of basis {e

}. Now let π : G → GL(V ) be a linear representation of a ﬁnite

group G in the vector space V . For each s ∈ G, we put

(s) = tr(π(s)).

The complex valued function χ

on G thus obtained is called the character of the represen-

tation π; the importance of this functio n comes primarily from the fact that it characterizes

the rep r ese ntation π.

Proposition 2.1.2. If χ i s the character of a representation π of degree n, we have:

1. χ(1) = n.

2. χ(s

−1

) = χ(s), for all s ∈ G

3. χ(tst

−1

) = χ(s), for all s, t ∈ G.

Proof. We have π( 1) = I, and tr(I) = n since V has dimension n; hence (1). For (2) we

observe th at π(s) h as ﬁnite order; consequently the same is true of its eigenvalues λ

, . . . , λ

and so these have absolute value equal to 1 (this is also a consequence of the fact that π

can be deﬁn e d by a unitary matrix. Thus

χ(s) = tr(π(s)) =



−1

= tr(π(s)

−1

) = tr

−1

(s) = tr

−1

Formula (3) can also be written χ(vu) = χ(uv), putting u = ts, v = t

−1

; hence it Follows

from the well known formula tr ( ab) = tr(ba), valid for two arbitrary linear mappings a and

b of V into itself.

A function f on G satisfying identity ( 3 ) above, is called a class function. We shall see later

that each class function is a linear combination of characters. We will use the followi n g

result [166, p.11 ]

Proposition 2.1.3. Let π

: G → GL(V

) and π

: G → GL(V

) be two linear representa-

tions of G, and let χ

and χ

be their characters. Then:

1. The character χ of the direct sum representation V

⊕ V

is equal to χ

+ χ

2. The character ψ of the tensor product representation V

⊗ V

is equal to χ

· χ

If φ and ψ are functions on G, set

φ, ψ =

|G|



g∈G

φ(g

−1

)ψ(g) =

|G|



g∈G

φ(g)ψ(g

−1

We have φ, ψ = ψ, φ. Moreover φ, ψ is lin ear in φ and in ψ. Consider the following

notation. If φ and ψ are two complex-valued functions on G, put

φ|ψ =

|G|



g∈G

φ(g)

ψ(g).

This is a s cal ar product; it is linear in φ, semilinear in ψ; and we have φ|φ > 0, for all

φ = 0. If

ψ is the function d eﬁned by the formula

ψ(g) =

ψ(g

−1

), th en we have

φ|ψ =

|G|



g∈G

φ(g)

ψ(g

−1

) = φ,

ψ.

In particular, if χ is the character of a representation of G, we h ave ˘χ = χ (Proposit io n

2.1.2), so that φ|χ = φ, χ, for all functions φ on G. So we can use at will φ|χ or φ, χ,

so long as we are concerned with characters. We have an important result on orthogonality

of characters [166, p.15]:

Theorem 2.1.4. 1. If χ is the character of an irreducible represent ation, we have χ|χ =

1 (i. e., χ i s of norm 1).

2. If χ and χ

′

are the characters of t wo nonisomorphic irreducible re presentations, we

have χ|χ

′

 = 0 (i. e., χ and χ

′

are orthogonal).

A character of an irreducib l e representation is called an irreducible character. Thus, The-

orem 2. 1. 4 shows that the irreducible characters form an orthonor m al system.

Theorem 2.1.5. Let V be a linear representation of G, with character φ, and suppose V

decomposes into a direct sum of irreducible representat ions:

V = W

⊕ W

⊕ ··· ⊕ W

Then, if W is an irreducible representation with character χ, the number of W

’s isomorphic

to W is equal to the scalar product φ|χ.

Proof. L et χ

be the charac t er of W

. By Proposition 2.1.3, we have φ = χ

+ . . . + χ

Thus, φ|χ = χ

|χ+ . . . + χ

|χ. But, accord ing to the preceding Theorem 2.1.4, χ

|χ

is equal to 1 or 0, depending on whether W

is, or is not, isomorph i c to W. The result

follows.

Corollary 2.1.6. The number of W

isomorphic to W does not depend on the chosen

decomposition.

This number is called the ’number of times that W occurs in V ’ , or the ‘number of times

that W is contained in V ’. It is in this se n se that on e can say that there is uniquen e ss in

the deco m position of a representation into irreducible representations.

Corollary 2.1.7. Two representations with the same character are isomorphic.

The above results reduce the study of repr esentations to that of their characters. If

, . . . , χ

are the distinct irreducible characters of G, an d if W

, . . . , W

denote corre-

sponding representations, each r ep r e sentation V of G is isomorphic to a direct sum

V = m

⊕ m

⊕ . . . ⊕ m

with m

, . . . , m

≥ 0 integers. The character φ of V is equal to m

+ . . . + m

and we

have m

= φ| χ

. We obtain thus a very convenient irreducibi li ty criterion [166, p.17]:

Theorem 2.1.8. If χ is the character of a representation V , φ|φ is a positi ve integer

and we have φ|φ = 1 if, and only if, V is irreducible.

Example 2.1.9 (The regular representation). Let n be the order of G, and let V be a

vector space of dimension n, wi t h a basis {e

}

g∈G

indexed by the elements g of G. For

s ∈ G, let π(s) be the linear map of V into V which sends e

→ e

; this d eﬁ n e s a linear

representation, which is called th e regular representation of G. Its degree is equal to the

order of G. Note that e

= π

); henc e note that the images of e

form a basis of V .

Conversely, let W be a re p r esentation of G containing a vector w such that the π

(w), g ∈ G,

form a basis of W ; then W is isomorphic to the regular repre sentation (an isomorphism

τ : V → W is deﬁned by putt ing τ (e

) = π

(w).

For the rest of this secti on, the irreduci b l e characters of G are denoted χ

, . . . , χ

; their

degrees are written n

, . . . , n

, where n

= χ

(1) (Proposition 2.1.2). Let R be the regular

representation of G. Recall that it has a basis {e

}

g∈G

such that π

) = e

. If s = 1,

we have sg = s for all g, which shows that the diagonal terms of the matrix of π

are zero;

in particular we have tr ( π

) = 0. On the other hand, for s = 1, we have tr(π

) = tr(I) =

dim R = n. Whence:

Proposition 2.1.10. The character r

of the regular representation is given by the for-

mulas:

(1) = n = |G|, r

(s) = 0, if s = 1.

Corollary 2.1.11. Every irreducible representation W

of G is contained in the regular

representation with multiplicity equal to its degree n

Proof. According to Theorem 2.1.4, this number is equal to r

|χ

, and we have

r

|χ

 =

|n|



g∈G

−1

)χ

(s) =



(1)χ

(1)





n · χ

(1)



= χ

(1) = n

Corollary 2.1.12. 1. The degrees n

satisfy the relation



= n.

2. If s ∈ G is diﬀerent f rom 1, we have n

(s) = 0.

Proof. By Corollary 2.1.11 , we have r

(s) =



(s), for all s ∈ G. Ta ki n g s = 1 we

obtain (1), and taking s = 1, we obtain (2).

Note that the above result can be used in determining th e irreducible representations of

a gro u p G: suppose we have constructed some mutually non isomorphic irreducible repre-

sentations of degrees n

, . . . , n

; in order that they be all the irreducible representations of

G (u p to isomorphism ) , it is necessary and suﬃcient that n

+ . . . + n

= n. Also, we will

see below that the degrees n

divide the order n = |G|.

Recall that a function f on G is called a class function if f(tst

−1

) = f (s) for all s, t ∈ G.

Denote by H the space of class funct i on s on G. In particular, the irreduci b l e characters

, . . . , χ

belong to H. In fa ct , [166, p.19]

Theorem 2.1.13. The irreducible characters χ

, . . . , χ

form an orthonormal basis of H.

Recall that two el em ents g and g

′

of G are said to be conjugate if there exists s ∈ G su ch

that g

′

= sgs

−1

; this is an equivalence re la t io n , which partitions G into classes (al so called

conjugacy classes). We have the following (see [166, p.19–20])

Theorem 2.1.14. The number of irreducible representations of G (up to isomorphism) is

equal to the number of conjugacy classes of G.

Proposition 2.1.15. Let s ∈ G, and let c(s) be the number of elements in the conju-

gacy class of s. Then,



(s)χ

(s) =

c(s)

, and for g ∈ G not conjugate to s, we have



(s)χ

(g) = 0.

We now give an exam p l e of a character table of a ﬁnite group.

Example 2.1.16 (Table of characters of S

). Take for G the group of symmetric grou p

of three elements S

. We have n = 6, and there are three conjugacy classes: the element

1 = (1); the three transpositions (1 2), (2 3), ( 3 1); and the two cyclic permutations (1 2 3),

(1 3 2). Let t be a transposition and c a cyclic permutation. We have t

= 1, c

= 1, and

tc = c

t (equivalently, ctct = 1). Whence th er e are just two characters of degree 1: the

unit character χ

and the character χ

giving the sig n at u r e of a pe rmutation (that is 1 for

the even per mutations and −1 for the odd ones). Theorem 2.1.14 above shows that there

exists one other irredu ci b l e character χ

; if n is its degree we must have 1 + 1 + n

= 6,

hence n = 2.

The values of χ

can be deduced from the fact that χ

+ χ

+ 2χ

is the character of t h e

regular representation r

of S

(Proposition 2.1.10 ) . We thus get the character table of

character 1 t c

1 1 1

1 -1 1

2 0 -1

Table 2.1: Charac te r table for the symmetric group S

2.2 Lie groups

Consider the general linear group over the real numbers, denoted GL

(R), i. e., the group of

all n ×n invertible matrices with real entries. Similarly, we consider the the general linear

group over the complex numbers of all n × n invertible matrices with complex entries,

denoted GL

(C). The gen er al linear groups are indeed groups under the operation of

matrix multiplication: the product of two invertible matrices is invertible, the identity

matrix is an identity for the grou p , an invertible matrix has (by d eﬁnition) an inverse, and

matrix multiplication is associative.

Let C

n×n

denote the space of all n × n matrices with com plex entries, and let {A

}

be a

sequence of complex matrices in C

n×n

. We say that {A

}

converges to a matrix A if each

entry of (A

)

of A

converges to the corresponding entry of A

of A.

Deﬁnition 2.2.1. A matrix Lie group is any subgroup G of GL

property: If {A

}

is any sequence of mat r i ces in G, and {A

} converges to some m at r i x

A then either A ∈ G, or A is not invertible.

The condition on G amounts to saying that G is a closed subgroup of GL

it does not necessarily mean that G is closed as a subset of C

n×n

. Most of the ‘interesting’

subgroups of GL

Example 2.2.2. The general linear gr ou ps GL

(R) and GL

Lie gro u p s.

Example 2.2.3. (The special linear gr oups SL

(R) and SL

(C))

The special linear group (over R or C) is the group of n × n invert ible m at r i ces (wit h rea l

or complex entries) having determin ant one. That is SL

Both of these are subgroups of GL

(C). Furthermore, if {A

}

is a sequence of matrices

with determinant one an d {A

} converges to A, then continuity of determinant fun ct io n

implies that A also has determinant one.

Example 2.2.4. (The or t hogonal and special orthogonal groups O

(R) an d SO

(R))

An n × n real matrix A is said to be orthogonal if the col u m n vectors that make up A are

orthonormal, that is, if



ℓ=1

ℓj

ℓk

= δ

, for all 1 < j, k < n.

Equivalently, A is orthogonal if it preserves the inner product, namely if x, y = Ax, Ay,

for all vectors x, y ∈ R

. Since det A

= d et A, we see that if A is orthogonal, then

(det A)

= d et( A

A) = det I = 1. Hence, det A = ±1, for all orthogonal matrices A.

In particu l ar , every orthogonal matrix must be invertible. However, if A is an ort hogonal

matrix, then

A

−1

x, A

−1

y = A(A

−1

)x, A(A

−1

)y = x, y.

Thus, the inverse of an orthogon a l matrix is orthogonal. Furthermore, the product of two

orthogonal matrices is o r t h ogo n al , since if A and B both preserve inner products, then so

does AB. Thus, the set of orthogonal matrices forms a group.

The set of all n × n real orthogonal matrices is the orthogonal group O

(R), and i t is a

subgroup of GL

(C). The limit of a sequence of orthogonal matrices is orthogonal, because

the rela t io n A

A = I is preserved under taking l i m i t s. Thus, O

(R) is a matrix Lie group.

The set of n ×n ort hogonal matrices with determinant one is the special orthogonal group

(R). This is also a matrix Lie group. Sometimes O

(R) and SO

(R) are si m p l y de-

noted by O(n) and SO( n) , respect i vely.

Example 2.2.5. (The complex orthog on a l groups O

(C))

Similarly to the example above, we can deﬁne orthogonal groups of complex matrices.

Consider the bilinear form ·, · on C

given by x, y =



. We deﬁne the complex

orthonormal group as

The complex special orthogonal group SO

A ∈ O

Example 2.2.6. (The u nitary group s U(n) and SU(n))

An n×n complex matrix A is said to be unitary if the column vectors of A are orthonormal,

that i s, if



ℓ=1

ℓj

ℓk

= δ

, for all 1 < j, k < n.

Equivalently, A is unit a r y if it preserves the inner pro d uct x, y



¯y

, namely if

Ax, Ay

= x, y

for all vectors x, y ∈ C

. Still another equivalent deﬁniti on i s tha t A is

unitary if A

∗

A = I, where, A

∗

is the adjoint (herm i t i an ) of A. Since det A

∗

= det A,

we see that if A is unitary, then det(A

∗

A) = |det A|

= 1. Hence, |det A| = 1, for all

unitary matrices A. In particu la r , this shows that every unitary matrix is invertible. The

same argument as for the orthogonal group shows tha t the set of unitary matrices forms a

group.

The set of all n × n unitary matrices is the unitary group U(n), and it is a subgroup o f

(C). The limit of unitary matrices i s unitary, so U(n) is a matrix Lie gr ou p. The set

of u n i t ar y matrices with determin a nt one is the special unitary group SU(n), a n d it can

be p r oved that SU(n) is a m at r i x Lie group.

Example 2.2.7. (The sy m p l ect i c groups Sp

(R), Sp

(C), an d Sp(n))

Consider the skew-symmetric bilinear form [·, ·] on R

deﬁned as follows:

[x, y] =



k=1

n+k

− x

n+k

). (2.1)

The set of all 2n × 2n matrices A which preserve [ · , ·] i s called the real symplectic group

(R), and it is a subgroup of GL

(C). In fact, Sp

(R) is a matrix Lie group. This

group arises natu r al l y in the study of classical m e chanics: If J is the 2n × 2n matrix

J =



0 I

−I 0



then [x, y] =  x, Jy, and it is possib l e t o check that a 2n ×2n real m at r ix A is i n Sp

(R) if

and on ly if A

JA = J. Taking the determinant of this identity gives (det A)

det J = det J,

or (det A)

= 1. This shows that d et A = ±1, for all A ∈ Sp

(R). In fact, det A = 1 for all

A ∈ Sp

(R).

One can deﬁne a bilinear form on C

by the same formula (2.1), since this form involves

no complex conjugates. The set of 2n × 2n complex matrices which preserve this form i s

called the complex symplectic group Sp

(C). Finally, we have the compact symplectic group

Sp(n) deﬁned as

Sp(n) = Sp

Example 2.2.8. (The Hei senberg group H)

The set of all 3 × 3 real matrices A of the form





1 a b

0 1 c

0 0 1





, wh er e a, b, c ∈ R,

is called the Heisenberg group H. In fact, H is a matrix Lie group. The reason for the

name Heisenberg group is that the Lie algebra of H gives a realization of the Heisenberg

commutation relations of quantum mechanics.

Example 2. 2.9 . The groups R

, C

, S

, R, and R

can be thought as subgroups of matri-

ces. For example, the group R

of non-zero real numbers under multiplication is isom o rphic

to GL

(R). Thus, we will regard R

as a matr i x Lie group . Similarly, the group C

. The

group S

of compl ex numbers module one is isomorphic to U(1).

The additive grou p R is isomorp h i c to GL

(R)

(1 ×1 real matrices with positive determi-

nant) vi a the map x → [e

]. In a similar way, the additive group R

is isom or phic to the

group of diagonal real matrices with positive diagonal entries, via the map

, . . . , x

) →



0 e



Now we discuss some notions of compact n ess and connectedness for matrix Lie groups.

Deﬁnition 2.2.10. A matrix Lie group G is said to be compact if the following two

conditions are sat is ﬁ ed :

1. [closeness] If {A

}

is any sequence of matrices i n G and {A

}

converges to a matrix

A, then A is in G.

2. [boundedness] There exists a constant C such that for all A ∈ G, |A

| < C for all

1 < i, j < n.

This is not the usual t opological deﬁnition of compactness. Thinking the set of all n × n

complex matrices as C

, the above deﬁnition says that G is compact if it is a closed,

bounded subset of C

. It is a stand ar d theo r em from eleme ntary analy si s th a t a subset

of C

is compact if and only if it is closed and bounded. All of our examples of matrix

Lie groups except GL

(R) and GL

condition (2) t h at is most important.

The groups O

orthogonal matrices is orthogonal and the limit of matrices with determinant one has de-

terminant one. Pr operty (2) is satisﬁed because if A is orthogonal, then the column vectors

of A have norm one, and hence |A

| < 1, for all 1 < i, j < n. A similar argument shows

that U(n), SU(n), and Sp(n) are compact , this includes th e unit circle, S

∼

U(1). On

the oth er hand, the groups GL

(R) and GL

matrices could be noninvertible. The grou p s SL

(R) and SL

except in the trivial case n = 1. The following groups also violate (2), and hence are

noncompact: O

(C); the Heisenberg group H; Sp

(R) and Sp

(C); R, C, R

and C

Deﬁnition 2.2.11. A matr i x Lie group G is said to be connected if given any two matrices

A and B in G, there exi st s a continuous path γ : [0, 1] → G, with γ(0) = A and γ(1) = B.

This property is what is called path-connected in topology, which is not (in gener a l) t he

same as connected. However, it is a fact that a mat r i x Lie grou p is connect ed if and

only if it is pat h - con nected. A matrix Lie group G which is n ot connected ca n be decom -

posed (uniquely) as a un i on of several pieces, called components, such that two el em ents

of the same component can be joined by a continuous path, but two elements of di ﬀe re nt

components cannot.

It is a known result that the general linear g ro u p GL

the group GL

(R) is not. In Table 2.2 we list the examples of matrix Lie grou p s a bove,

indicating their connectedness properties.

Group Connected? Number of Components

(R) no 2

(R) yes 1

(R) no 2

(R) yes 1

U(n) yes 1

SU(n) yes 1

Heisenberg yes 1

Table 2.2: Conne ct ed n es s propert i es of some classical matrix Lie gr oups.

In a similar manner, we can st u dy the notion of simply connected for mat r i x Lie groups:

Deﬁnition 2.2.12. A matrix Lie group G is sai d t o be simply connected if it is connected

and, in addition, every loop in G can be shrunk continuously to a point in G. More precisely,

if gi ven any continuous path γ : [0, 1] → G, with A(0) = A( 1) , there exists a continuous

function H(s, t) : [0, 1] × [0, 1] → G having the foll owing properties:

(a) H(s, 0) = H(s, 1) for all s ∈ [0, 1].

(b) H(0, t) = γ( t), for all t ∈ [0, 1].

Table 2.3 resumes the examples o m at r i x Lie groups above, indicat i n g their simply con-

nectedness properties.

We conclude this section with the deﬁnition of a Lie group:

Deﬁnition 2.2.13. A Lie group is a diﬀerentiable manifold G which is also a group and

such tha t the group product

G × G → G

Group Simply connected? Fundamental group

(R) no Z

(R), n ≥ 3 no 2Z

U(n) no Z

SU(n) yes {1}

Sp(n) yes {1}

(R)

, n ≥ 2 no same as SO

(R)

(R) no same as SO

(R)

(R) no Z

Table 2.3: Sim p l y connectedne ss propert i es of some classical matrix Lie gr oups.

and the inverse map g → g

−1

are diﬀer entiable.

A manifold is an object that looks locally like a p i ece of R

, that is, a topological space that

is Hausdorﬀ, second cou ntable, and locally Eucl i d ean. An example would be a torus. For

a precise deﬁnition, see [128] or [129]. We conclude with a result that establishes whether

every matr i x Lie group is a Lie group [91, p.52].

Theorem 2.2.14. Every matrix Lie group is a smooth embedded submanifold of C

n×n

and

is thus a Lie group.

2.3 Lie algebras

A nonassociative algebra is a vector space A (over a ﬁeld F of characteristic 0) equipped

with a bilinear map, called product from A × A to A. The algebr a A is called associative

if it contains an identity element 1 for multiplication, so that

1a = a = a1, ∀a ∈ A,

and i f the associative law a(bc) = (ab)c holds, for all a, b, c ∈ A. The algebra A is said to

be commutative if the commutative law ab = ba holds, ∀a, b ∈ A.

For subspaces B, C of an a lg eb r a A, we write BC for the subspace of A spann ed by the

products bc, for b ∈ B, c ∈ C. Similarly, we denote by [B, C] the subspace BC − CB =

{[b, c] : b ∈ B, c ∈ C} spanned by al l th e commutators [b, c]. A subalgebra of A is a subspace

B of A such that B

⊆ B, and 1 ∈ B in th e associative case. Equivalently, a subalgebra of

A is a subset B of A which is an algebra under the linear and product structures induced

from A.

For algebras A and B, a lin ear map f : A → B is an homomorphism if f(ab) = f( a) f (b),

for a, b ∈ A, and if i n addition f(1) = 1 in the as sociative case. The homomorph i sm f is

an isomorphism if it is a li near isomorphism . In case A = B, an homomorphism is called

an endomorphism, and an isom o r p h i sm is called an automorphism. Two algebras A and B

are said isomorphic if there is an isomorphism f : A → B. In thi s case, we write A

∼

A linear endom or phism d : A → A of an algebra A is called a derivation if for all x, y ∈ A

we have d(xy) = d(x)y + xd(y).

Deﬁnition 2.3.1. A Lie algebra is a nonassociat ive algebra g such that:

(i) ( al t er n at e axiom) [x, x] = 0, ∀x ∈ g.

(ii) (Jacobi’s id entity) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0, ∀x, y, z ∈ g.

The alternate pr operty is equivalent to the skew-symmetry condition [x, y] + [y, x] = 0,

∀x, y ∈ g. The Jacobi identity is equivalent to the condition that the adjoint map

ad x : g → g, given by ad x(y) = [x, y]

be a derivation, for all x ∈ g.

A Lie algebra g is called commutative or abelian if [g, g] = 0. Two elements x, y of a L ie

algebra g are said to commute if [x, y] = 0. In particular, all pairs of elements of an abelian

Lie algebra commute. For Lie algebras g and k, a linear map f : g → k is an homomorphism

if f is an algebra homomorphism and it sat i sﬁ es

f([x, y]) = [f(x), f(y)], ∀x, y ∈ g.

We deﬁne in obvious way t h e term isomorphism, endomorph i sm and automorphism.

Example 2.3.2. An associative algebra is a Lie algeb r a wit h t h e br acket [x, y] = xy −yx.

Example 2.3. 3. Let B be a nonassociative algeb r a , and set A = End B. Then, the space

of deri va ti o n s of B forms a Lie subal gebra of A.

Example 2.3.4. Every one- d i m ensional Lie algebra is abelian.

Lie algeb r as are simpler than matrix Lie groups, because (as we have seen) the Lie algebra

is a linear space. Thus, we can understand much about Lie algebras just by doing linea r

algebra. In fa ct , to each matrix Lie grou p G we can associat e a L i e algebra, just by

considering the expon en ci a l of a matrix. Let X be an n × n real or complex m at r i x. We

deﬁne t h e exponenti al matrix of X, d en ot e d e

or exp X, by the usual power series



m≥0

. (2.2)

We know from the diﬀerential equation th e or y that th e exponencial series of any ma tr i x

A ∈ GL

the homogeneous linear equation given by X

′

= AX, X(0) = I (see [168] for a proof).

Also, t h e exponen cia l satisﬁes the usual properties:

= Ae

;

2. e

(s+t)A

= e

, for all s, t ∈ R;

3. e



m≥0

;

4. BC = CA impli es e

C = Ce

;

5. e

t(A+B)

= e

if and only if AB = BA. In particular , (e

)

−1

= e

−tA

;

for all A, B, C ∈ GL

Deﬁnition 2.3.5. Let G be a matr i x Lie group in C

n×n

. The Lie algebra of G, denoted

by g, is the set of all matrices X ∈ C

n×n

such tha t e

is in G for all real numbers t.

This means that X is in g if and only if the one-parameter subgroup generated by X lies

in G. Note that even though G i s a subgroup of GL

(R)),

we do not require that e

be in G for all complex numbers t, but only for all real numbers

t. Also, it is not enou gh to have just e

∈ G. It can be given an exam p l e of an X an d a

G such that e

∈ G, but such that e

/∈ G for some real va l u es of t. Such an X is no t in

the Li e algebra of G.

On the other han d , since all matrix Lie group G is itself a Lie group, and thus a smooth

manifold (Theorem 2.2 .1 4), we can obtain the tan gent space of G on the point I, the iden-

tity matrix of G, namely T

G. It is a well kn own result that, if X is on T

G, then e

∈ G,

for all t ∈ R (see [134] for a proof).

For A, B ∈ T

G, we deﬁne the bracket [ · , ·] by [A, B] = AB − BA. Since we ca n write

√

= I +

√

tA +

+ o(t),

√

= I +

√

tB +

+ o(t),

−

√

= I −

√

tA +

+ o(t),

−

√

= I −

√

tB +

+ o(t),

and set λ(t) = e

√

−

√

−

√

, then we have

λ(t) = e

√

−

√

−

√

= I +

√



(A + B) − ( A + B)



+ t



+ 2AB + B

− (A + B)



+ o(t)

= I + t[A, B] + o(t),

with lim

t→0

o(t)

= 0. Th en , λ

′

(0) = [A, B]. Since λ(t) ∈ G, for all t ∈ R, we see that

[A, B] ∈ T

G. We have also proved that T

G admits a Lie algebra structure. In fact, th e

Lie alg eb r a g of a matrix L ie group G is given by

g = T

and it directly follows that T

G = T

G · X, for a ny X ∈ G (see [129 ] or [134]).

Note that we can a lso see the Lie algeb r a g of a Lie group G as the set of all left-invariant

vector ﬁelds X : G → T G (or linear derivations X : C

∞

(G) → R), wi t h some bracket

[X, Y ] (see [91 ]) . Because g is a real sub al gebra of th e space GL

results, [91, p. 55] and [179, p.237]

Proposition 2.3.6. The Lie algebra g associated to the Lie group G is a real Lie algebra.

Theorem 2.3.7 (Ad o ) . Every ﬁnite-dimensional real L ie algebra is isomorphic to a subal-

gebra of gl

(R). Every ﬁnite-dimensional complex Lie algebra is isomorphic to a subalgebra

of gl

(C).

We present some examples of Lie algebras associated to some Lie groups st u died in previous

section:

Example 2.3.8 (The g en er al linear groups). If X is any n × n complex matrix, then e

is invertible. Thus, the Lie al geb r a of GL

This Lie algebra is denoted gl

(C). If X is any n×n real mat r i x, th en e

will be invertible

and real. On the other hand, if e

is r eal for all real numbers t, then X =







t=0

will

also be real. Thus, the Lie algebra of GL

(R) is the space of all n × n real matrices, and

is den ot ed gl

(R).

Example 2.3.9 (The special linear groups). It is a well known restult that det e

= e

tr X

Thus, if tr X = 0, then det e

= 1 for all real numbers t. On the other hand, if X is any

n × n matrix such that det e

= 1 for all t, then e

t tr X

= 1 for all t. This means that

t tr X is an integer multiple of 2πi for all t, which is only possi b l e if tr X = 0. Thus, the

Lie algebra of SL

(C).

Similarly, the Lie algebra of SL

(R) is the space of all n ×n real matrices with trace zero,

denoted sl

(R).

Example 2.3.10 (The unitary groups). Recall that a matr i x U is unitary if and only if

∗

= U

−1

. Thus, e

is un it a r y if and only if

)

∗

= (e

)

−1

= e

−tX

. (2.3)

Since (e

)

∗

= e

∗

, th en (2.3) becomes e

∗

= e

−tX

. Clearly, a su ﬃcient condition for th i s

last identity to hold is that X

∗

= −X. On the o ther hand, if e

∗

= e

−tX

holds for all t,

then by diﬀerentiating at t = 0, we see that X

∗

= −X is necessary. Thus, the Lie al gebra

of U(n) is the space of all n × n complex matrices X such that X

∗

= −X, and is denoted

by u(n).

Combining the two previous computations, we see that the Lie algebra of SU(n) is the

space of all n × n complex matrices X such that X

∗

= −X and tr X = 0, denoted su(n).

Example 2.3.11 (The orthogonal groups). The identity component of O(n) is just SO(n).

Since the exponential of a matrix in the Lie algebra is automatically in the i d entity com-

ponent, the Lie algebra of O(n) is the same as the Lie algebra of SO(n). Now, an n × n

real mat r ix R is orthogonal if and only if R

= R

−1

. So, given an n × n real matrix X,

is orth og on al if and only if (e

)

= (e

)

−1

, or

= e

−tX

. (2.4)

Clearly, a suﬃcient condition for th i s to hold is that X

= −X. If (2.4) holds for all

t, then by diﬀerentiating at t = 0, we must have X

= −X. Thus, the Lie alg eb r a of

O(n), as well as the Lie algebra of SO(n), is the space of all n × n real matrices X with

= −X, denoted by so

(R) (or simply so(n)). Note that t he condition X

= −X forces

the di ag on al entries of X to be zero, and so, n ecess ar i ly the trace of X is zero.

The same argument shows that the Li e algebra of SO

matrices satisfying X

= −X, denoted by so

(C). Th i s is not the same as su(n).

Example 2.3.12 (The symplectic groups). These are d en o te d sp

(R), sp

(C), and sp(n) .

The calculation of t hese Lie algebras is sim i l ar to that of the generalized orthogonal group s,

and we will just record the result h er e. Let J be the matrix in the deﬁnition of the symplectic

groups. Then, sp

(R) is the space of 2n × 2n real matrices X such th at JX

J = X,

Example 2.3.1 3 (The Heisenb er g group). Recall that the Heisenberg group H is the group

of all 3 × 3 real matrices A of the form





1 a b

0 1 c

0 0 1





, wh er e a, b, c ∈ R.

Computing the exponential of any matrix of the form

X =





0 α β

0 0 γ

0 0 0





, wh er e α, β, γ ∈ R, (2.5)

we can see that e

is i n H. On the other hand, if X is any matrix such that e

is i n H

for all t, th en all of the entries of X =



t=0

which are on or below the diagonal must

be zero, so th at X is o f form (2.5) . Thus, the Lie algebra of the Heisenberg gr ou p is the

space of all 3 × 3 real matrices that are strictly upper triangular.

Deﬁnition 2.3.14. If V is a ﬁni t e- d i m en si onal real vector space, then t h e complexiﬁcation

of V , den ot ed V

, is the space of formal linear combinations

+ iv

, wit h v

, v

∈ V.

This becomes a real vector space in the obvious way and becomes a complex vector space if

we deﬁ ne i(v

+ iv

) = −v

+ iv

. We will regard V as a real subspace of V

in the obvious

way. Now let g be a ﬁnite-dimension al real Lie algebra and g

its com p l ex iﬁcation (a s a

real vector space). Then, the bracket operation on g has a unique extension to g

which

makes g

into a complex Lie algebra. In fact, the uniqueness of the extens io n is obvious,

since if the bracket operation on g

is to be bilinear, then it must be gi ven by

+ ix

, y

+ iy

] =



, y

] − [x

, y

]



+ i



, y

] + [x

, y

]



. (2.6)

This bracket deﬁned above is really bilinear and skew symmetric and it s at i sﬁ es the Jacob i

identity. It is clear from (2.6 ) is re al b i l inear, and skew-symmetric. The skew symmetry

means that if (2.6) is complex linear in the ﬁrst factor, it is also complex linear in th e

second fact or . Thus, we need only show that

[i(x

+ ix

), y

+ iy

] = i[x

+ ix

, y

+ iy

]. (2.7)

The left-hand side of (2.7) is [−x

+ix

, y

+iy

] = (−[x

, y

]−[x

, y

])+i([x

, y

]+[x

, y

]),

whereas t h e right-hand side of (2.7) is just



, y

] − [x

, y

]



+ i



, y

] + [x

, y

]



= (−[x

, y

] − [x

, y

]) + i( [x

, y

] + [x

, y

]),

and, in d ee d , these are equal. To check the Jacobi identity, note that the Jacobi identity

holds if x, y, and z are in g. However, observe that t h e expression on the left-hand side of

the Jacobi identity in Deﬁnition 2.3.1 is (complex!) linear in x for ﬁxed y an d z. It follows

that the Jacobi identity holds if x is in g

, and y and z are in g. The same argum e nt shows

that we can extend to y i n g

, a n d then to z in g

. Thus, the Jacobi identity holds i n g

Thus, we have

Proposition 2.3.15. Let g be a ﬁnite-dimensional real Lie algebra and g

its complexiﬁ-

cation (as a real vector space). Then, the bracket operation on g has a unique extension to

which makes g

into a complex Lie algebra.

Deﬁnition 2.3.16. The complex Lie algebra g

is called the complexiﬁcation of the real

Lie alg eb r a g.

We give the complexiﬁcations of some real Li e algebras in Table 2.4

Lie algeb r a Complexiﬁcation

(R) gl

(R)

= gl

(C)

u(n) u(n)

= gl

(C)

su(n) su(n)

= sl

(C)

(R) sl

(R)

= sl

(C)

so(n) so(n)

= so

(C)

(R) sp

(R)

= sp

(C)

sp(n) sp(n)

= sp

(C)

Table 2.4: Compl ex iﬁcations of some classical real Lie algebras.

2.4 Mod ul es

Deﬁnition 2.4.1. Let A be an associative algebra and let V be a vector sp a ce. We s ay

that V is an A-module if there is a bilinear map A×V → V (denoted by a dot (a, v) → a·v)

such tha t

(i) [ identity] 1 · v = v, for all v ∈ V .

(ii) [associativity] (ab) · v = a · (b · v), for all a, b ∈ A, and all v ∈ V .

For a ∈ A, let π

the correspond i n g linear endomorphism of V , such that π

(v) = a · v.

Then, t h e map

π : A → End V, such that π(a) = π

is an homomorp h i sm of associative algebras. Such an homomorphism is called a represen-

tation of A on V . Sometim es, V is called a representation of A.

Example 2.4.2. Note that the associative algebra A ha s a natural representation on itself,

given by the left multiplication action: a · b = ab, ∀a, b ∈ A.

Deﬁnition 2.4.3. Analogously, let g be a Lie algebra and let V be a vector space. Then V

is called a g-module if there is a b i l i n ear map g ×V → V (denoted by a dot (x, v) → x ·v)

such tha t

(i) ( b r acket preserving) [x, y] · v = x · (y · v) − y · (x · v), for all x, y ∈ g, an d all v ∈ V .

If for x ∈ g, we denote π

the corr esponding linear endomorphism of V , such that

(v) = x · v.

Then, t h e map

π : g → End V, such that π(x) = π

is a Lie algebra homomorp h i sm . Such an h o m om o rphism is called a representation of g on V

(sometimes V is called a representation of g). The notio n s of g- m odule and representation

of g are equivalent.

The Lie algebra g has a natural representation on itself, called the adjoint representation,

by the map

ad : g → End g, such that x → ad x.

In p ar t i cu l a r, taking g = End V , every Lie subalgebra of End V has a natural representa-

tion on V .

Let g be an associative or Lie algebr a , and let V be a g-module. For subspaces h of g and

W of V , we denote h · W the linear span of all x · w, for x ∈ h, w ∈ W . A submodule of V

is a sub space W of V such that g · W ⊆ W , or equivalently, a subset W of V which is a

g-module under the linear stru ct ure a n d g-module action induced from V . A subsp ac e W

of V i s invariant (under g) if it is a su b m odule. The module V is said irreducible or simple

if V = 0 and if V has no proper nonzero invariant subspaces. The module V is called

indecomposable if it cannot be decomposed as a direct sum of two nonzero submo d u l es.

Clearly, an irreduci b l e module is indecomposable. Let V and W be g-modules. A linear

map f : V → W is called a g-module homomorphism or g-module map if

f(x · v) = x · f(v), ∀x ∈ g, ∀v ∈ V.

Such a map f is called a g-module isomorphism or g-module equivalence if it is a linear iso-

morphism. Two modul es V and W are isomorphic or equivalent if there is an isomorphism

f : V → W . In that case we write V

∼

W .

Deﬁnition 2.4.4. A subspace a of a Lie algebra g is called an ideal of g if [g, a] ⊆ a.

Equivalently, an i d eal of g is a subm odule under the adjoint representation.

An ideal is a su b al ge b r a. Given an ideal a of g, the quotient vector space g/a becomes a

Lie algebra, called the quotient Lie algebra, by means of the well deﬁned nonassocia t ive

product

[x + a, y + a] = [x, y] + a, ∀x, y ∈ g.

The canonical map π : g → g/a is an homomorphism, and we have an exact sequence of

Lie alg eb r as

0 −→ a −→ g

−→ g/a −→ 0.

A Lie algebra g is said to be simple if g is non zer o and has no proper nonzero ideals (equiv-

alently, the adjoint representation is simple), and if dim g > 1 (i. e., g is not abelian) .

Remember that a sub sp a ce I of an associative algebra A is called a left ideal (respectively

right ideal) of A if AI ⊆ I (respectively IA ⊆ A), and an ideal if is both left and right ideal.

Since an i d eal I need not to contain 1, it need not be a subalgebra. Given and ideal I of

A, the qu ot i ent vector space A/I becomes an associative algebra —the quotient algebra—

in an obvious way.

Example 2.4.5. The kernel of any homomorp hism of a Lie (respectively associative) al-

gebra g is an ideal.

In particular, the kernel of the adjoint representation of a Lie algebra g is an important

ideal c al le d the center of g and de n ot ed by Cent g:

Cent g = {x ∈ g : [x, g] = 0}.

Any subs p ace of Cent g is an ideal in g and is said to be a central ideal.

Deﬁnition 2.4.6. Given the Lie algebras a and b, an extension of a by b is a Lie algebra

g together with an exact sequence

0 −→ b −→ g −→ a −→ 0,

(note that b is an ideal and g/b

∼

a). This extension is said to be central if b is a central

ideal o f g.

Two extensions g

and g

of a by b are equivalent if there is an i so m or p h i sm g

∼

making the following diagram commute

∼











If a and b are ideals of a Lie algebra g, then a + b, a ∩ b and [a, b] are ideals also. In

particular, [g, g] is an ideal of g, called the commutator ideal, and is denoted by

′

= [g, g].

Given two Lie algebras a and b, their direct product is the Lie alge b r a a ×b, which is a ⊕b

as a vector space, with a and b retaining their original bracket structures and commuting

with one another. In particular, a and b are ideals of a × b. We deﬁne the di r ect product

of ﬁni t el y many Lie algebras analogously.

More generally, suppose that we have a representation π : a → End b of a Lie algebra a on

a Lie algebra b by derivations, i. e., π(x) = π

is a derivation of b, for all x ∈ a. Then, the

vector sp a ce a ⊕b carries a unique Lie algebra structure such that a and b are subalgebras

and [x, y] = π

(y), for all x ∈ a, y ∈ b (observe that b is an ideal, but n o t necessarily a).

This Lie algebra is called the semidirect product of a and b, and we denote this by a ⋉ b. A

Lie algebra is a semidirect product whenever it i s a vector space direct sum of a subalgebra

and a ideal. In fact, a ⋉ b = a × b if and only i f π ≡ 0. A semidirect pr oduct a ⋉ b is an

extension of a by b. An extension of a Lie algebra a by a Lie algebra b is said trivi al if it

is equi valent to a semidirect product a ⋉ b.

Example 2.4.7. We mention a particular kind of semidirect product. Given a L i e algebra

g and a derivation d of g, we can form Fd ⋉ g. This procedure is called adjoining the

derivation d to g.

Deﬁnition 2.4.8. Let S be a set. A vector space V is sai d to be S-graded if it is the direct

sum

V =



α∈S

of subsp aces V

. In this case, the elements o f V

are said to be homogeneous of degree α,

and V

is called the homogeneous subspace of degree α or the α-graded component of V .

For v ∈ V

(including v = 0) we write

deg v = α.

Given another S-graded vector space W , a linear map f : V → W is called grading

preserving if

f : V

→ W

, for all α ∈ S.

If such a map i s a linear isomorph is m , V and W are said to be graded-isomorphic. If S is

an abelian group, a linear map f : V → W is said to be homogeneous of degree β if

f : V

→ W

α+β

, for all α ∈ S. (2.8)

In that case, we write deg f = β. Not e that f is grading-pr es er vi n g if and only if deg f = 0.

If S is an abel ia n group which is a subgroup of the add i t i ve group of F, we can deﬁne the

degree operator d : V → W by th e condition

d(v) = αv, for v ∈ V

, α ∈ S. (2.9)

Note that a linear map f : V → W is grad i n g- p r es er vi n g if and on l y if [d, f ] = 0, and is

homogeneous of degr ee β if and only if [d, f] = βf.

A subspace W of a S-graded vector space V is graded if W =



α∈S

, where W

W ∩ V

, for α ∈ S. In this case, the quotient V /W is graded in a natural way:

V/W =



α∈S

(V/W )



α∈S

Given a family of S-graded vector spaces {V

}

, the direct sum X =



is natural ly

S-graded if we t a ke



, for α ∈ S.

2.5 Tensor products

Deﬁnition 2.5.1. Let V be a vector space over a ﬁeld F wit h a basis {e

}

and let W be

a vector space over F with a basis {f

}

. The tensor product V ⊗W is a vector space over

F, with a basis {e

⊗ f

}

i,j

If x =



∈ V and y =



∈ W , we have a tensor multiplication deﬁned by

x ⊗ y =



i, j

⊗ f

) ∈ V ⊗ W.

The tensor map τ : V × W → V ⊗ W d eﬁned by (x, y) → x ⊗ y satisﬁes bilinearity:

(i) ( x + y) ⊗ z = (x ⊗ z) + (y ⊗ z);

(ii) x ⊗ (y + z) = (x ⊗ y) + (x ⊗ z);

(iii) (λx) ⊗ y = λ(x ⊗ y) = x ⊗ (λy).

Conversely, if β is a bil i n ea r mapping of V ×W into another vector space X, then β(x, y) =









i,j

β(e

, f

) by bilinearity, and the linear transformation T :

V ⊗W → X t h at sends e

⊗f

to β(e

, f

) also sends x ⊗y to β(x, y), for all x ∈ V, y ∈ W .

In fact, T is the unique linear tr an sfo rmation such that β = T ◦ τ. In thi s sense, every

bilinear mapping of V × W factors uniquely through β (universal property):

V × W



V ⊗ W

;;

Tensor products of modules over a commutative rin g R can be deﬁned by the same universal

property (which does not require bases). More explicitly, we have [88, p.434]

Proposition 2.5.2 (Universal property for tensor produ ct s) . Let R be a commutative ring

and let A, B, C be R-modules. For a mapping β : A × B → C the following conditions are

equivalent:

1. β is bilinear;

2. a → β(a, ·) is a module homomorphism of A into Hom

(B, C);

3. b → β(·, b) is a module homomorphism of B into Hom

(A, C).

Bilinear mappings can be deﬁned in th e same way for left R-modules over an arbitrary r i n g

R, but then l ose properties (2) and (3) above, if only because Hom

(A, C) and Hom

(B, C)

are only abelian g r ou p s . It is more fruitful to keep properties (2) and (3), and to forgot

bilinearity unless R is commutative. In the simplest form of ( 2 ) and (3), C is an abelian

group; i f B is a left R-module, then Hom

(B, C) is a right R-module and A needs to be a

right R-module; so B and Hom

(A, C) are left R-modul es. Also, [88, p.435]

Proposition 2.5.3. Let R be a ring, let A be a right R-module, let B be a left R-module,

and let C be an abelian group. For a mapping β : A ×B → C the following conditions are

equivalent:

1. For all a, a

′

∈ A, b, b

′

∈ B and r ∈ R

(i) [biadditive] β(a + a

′

, b) = β(a, b) + β(a

′

, b), β(a, b + b

′

) = β(a, b) + β(a, b

′

(ii) [balanced] β(ar, b) = β(a, rb);

2. a → β(a, ·) is a module homomorphism of A into Hom

(B, C);

3. b → β(·, b) is a module homomorphism of B into Hom

(A, C).

We say that a bihomomorphism of modules is a mapping that satisﬁes the equivalent

conditions in Proposition 2.5.3. For example, the left a ct i on (r, x) → rx of R on any left

R-module M is a bihomomorphism of R × M into the underlying abelian group M. If

β : A × B → C is a bihomomorphism and ϕ : C → D is an homomorp h i sm of abelian

groups, then ϕ ◦ β : A × B → D is a bihomomorph is m . The tensor product of A and

B is an abelian group A ⊗

B with a bihomomorphism τ of A × B, fr om which every

bihomomorphism of A ⊗ B can be recovered uniquely in this fashion:

A × B



A ⊗

Deﬁnition 2.5.4. Let A be a right R-module and let B be a left R-module. A tensor

product of A and B is an abelian group A ⊗

B together with a bihomomorphism τ :

A × B → A ⊗

B given by (a, b) → a ⊗ b, the tensor map, such t hat, for every abelian

group C and bihomomorph is m β : A × B → C there exists a unique homomorph is m

β : A ⊗

B of abeli an groups such that β =

β ◦ τ.

If S i s an abelian group and if V and W are S-graded vector spaces, then V ⊗W acquires

a uni que S-grading by t h e condition

⊗ W

⊆ (V ⊗ W )

α+β

, for α, β ∈ S.

Using t h e symbol d

for th e degree operator in the space U, we have

V ⊗W

= d

⊗ 1 + 1 ⊗ d

This tensor product grading extends to an arbitrary ﬁnite number of tensor factors.

Now, let G be an abelian group and let A be a nonassociative algebra. Then A is a G-graded

algebra if it is G-graded as a vector space, so that

A =



α∈G

and if A

⊆ A

α+β

, for α, β ∈ U.

2.6 Mod ul e constr ucti o n

Fix an associative Lie algebra g. Let V a g-module and U ⊆ V a submodule. Then the

quotient vector space V /U becomes a g-module, ca l led the quotient module, by means of

the well deﬁned action

x · (v + U) = x · v + U, for x ∈ g, v ∈ V.

We have an exact sequence of g- m odules

0 −→ U −→ V −→ V/U −→ 0.

Given two g-modules V

and V

, their direct sum V

⊕V

is the g-module which is V

⊕V

as a vector space, with V

and V

retaining their original modu l e structures. In particular,

and V

are submodules of V

⊕ V

. The direct sum of any collection {V

}

of g-m odules

is deﬁ n ed analogously and is denoted by



A g-module is called completely reducible or semisimple if it is a direct sum of irreducible

submodules (here the null sum is allowed, so that the zero-dim ensional module is considered

completely reducible). Let G an abelian group and suppose that g is G-graded. A g-module

V is G-graded if it is G-graded as a vector space, so that V =



α∈G

, and if

· V

⊆ V

α+β

, for α, β ∈ G,

i. e., g

acts as operators of d egr ee α (s ee (2.8)). Quotients and direct sums of G-g ra d ed

modules are graded (as modules). In case g i s a G-graded Lie algebra, with G a subgroup

of the additive group of F, let d be the degree derivation of g. Then a G-graded g-module

V becomes an Fd ⋉ g-module when d is required to act as t he degree operator (2.9) on V

(observe t h a t d plays two diﬀe re nt compatible roles).

The grading of a graded module can be shifted in the following sense. Suppose that G is a

subgroup of an abelian group B a n d that V is a G-graded A- m odule, A a G-graded nonas-

sociative alg eb r a. Let β ∈ B. Then for each α ∈ G, V

can be renamed V

α+β

, gi vi ng V the

structure of a B-graded m odule with A

= 0 for γ ∈ B −G and V

= 0 for γ ∈ B −(G + β).

Now let g be a Lie algebra. In p r eparation for constructing the tensor produ ct of g-mod u l es,

we ﬁrst note that if π

and π

are two representations of g on V which commute in the

sense th at

[π

(x), π

(y)] = 0, for all x, y ∈ g,

then π

+ π

is a representation of g on V .

Deﬁnition 2.6.1. Given two g-modules V and W , we deﬁne the tensor product module

V ⊗W to be the vector space V ⊗W wi t h the acti on of x ∈ g determin e d by the condition

x · (v ⊗ w) = ( x · v) ⊗ w + v ⊗ (x · w), for v ∈ V, w ∈ W.

This is a g-module action because the equations

x · (v × w) = ( x · v) ⊗ w and x · (v × w) = v ⊗ (x · w)

clearly deﬁne two commuting g-module structures on the vector space V ⊗W . The tensor

product of ﬁnitely many g-modules is deﬁned a n al og ou sl y. If th e tensor factors are G-graded

modules (G an abelian group), then so is the te n sor prod u ct .

2.7 Induced modules

Let B a subalgebra of an associative algebra A and let V be a B-mod u l e. We denote by

A ⊗

V the quotient of th e vector space A ⊗

V by the subspace sp an ned by the el em ents

ab ⊗ v − a ⊗ b · v, for a ∈ A, b ∈ B, v ∈ V , and we again write a ⊗ b for the image of

a ⊗ v ∈ A ⊗

V in A ⊗

V . The space A ⊗

V car ri es a natu r al A-m odule structure

determined by the condition

c · (a ⊗ v) = ca ⊗ v for a, c ∈ A, v ∈ V,

and A ⊗

V is called the A-module induced by the B-mod ule V . It is sometimes denoted

as Ind

V .

There is a canonical B-module map i : V → A ⊗

V , given by v → 1 ⊗ v, and Ind

V has

the foll owing universal property: Given any A-module W a n d B-m odule map j : V → W ,

there i s a unique A-module map f : Ind

V → W making the foll owing diagram commute

Ind

;;

This property characterizes the A- m odule Ind

V and the map i up to canonical isomor-

phism. In fact, if I

′

is another A-module with a B-module map i

′

: V → I satisfyin g the

same condition, the we obtain A-module maps f : Ind

V → I, g : I

′

→ Ind

V . But g ◦f

and the identity map both make the diagram

Ind

commute, so that g ◦f is th e identity map by u niqueness. Similarly, f ◦g is the identity on

′

. If the algebra A and subalgeb r a B are G- gr a d ed and if V is a G-graded B-module (G

an abelian g r ou p ) , t h en the induced module Ind

V is a G-graded A-module in a natural

way.

Given a group G we deﬁ ne its group algebra to be the associative algebra F[G], which is

formally the set of ﬁnite linear combinations of elements o f G. That is, F[G] has the set

G as a linear basis, and multiplication in F[G] is simply deﬁned by line ar extension of

multiplication in G. The identity element of F[G] is just the identity element of G.

Deﬁnition 2.7.1. A representation of the group G on a vector space V is a group homo-

morphism π : G → Aut V . Th e space V is called a G-module or representation of G, and

just as for associative and Lie algebras, we often use the dot notation

g · v = π(g)v for g ∈ G, v ∈ V.

We have 1 ·v = v and (gh) ·v = g ·(h·v), for all g, h ∈ G, and all v ∈ V . We have the usual

module-theoretic concepts such as irredu ci b i l ity and equivalence. If π(G) = 1, π i s called a

trivial representation. Given G-m odules V

, . . . , V

, their tensor product is the vector space

⊗ . . . ⊗ V

, with G-action determined by

g · (v

⊗ . . . ⊗ v

) = (g · v

) ⊗ . . . ⊗ (g · v

) for g ∈ G, v

∈ V

The group G has a natural representation on its own group algeb r a , given by the left mul-

tiplication action. This is called the left regular representati on of G.

Any G-module V becomes a F[G]-module in a canonical way —by extending the map

π : G → Aut V by linearity to an algebra homomorphism of F[G] to End V —. In fact,

the G-modules are essentially the same as the F[G]-modules. For exampl e, the left regula r

representation of G corresponds to the left multiplication representation of F[G]. If the

group G is an abelian group written additively, such as the group Z, there can be confusion

as to whether t he symbol a + b means the sum in G or t h e sum in F[G], for a, b ∈ G. For

this reason we use t he exponential notation for the elements of G viewed as elements of

F[G] when G is such a group: we write e

for the element of F[G] corresponding to a ∈ G.

In part i cu l a r, e

= 1 and e

= e

a+b

, for a, b ∈ G.

Given a subgroup H of a group G and a H-module V , we d eﬁ n e th e G-module induced by

V to be th e G-modul e associated with the induced F[G]-module F[G] ⊗

F[H]

V . We write

Ind

V = F[G] ⊗

F[H]

There is a canonical H-module map i : V → F[G] ⊗

F[H]

V given by v → 1 ⊗ v, and the

induced module is characterized by the following universal property: Given any G- m odule

W and H-module map j : V → W , there is a unique G-module map f : Ind

V → W such

that t h e diagram

Ind

;;

commutes. It is clear that if X ⊆ G contains exactly one element from each of the left

cosets gH of H in G, the we have a linear isomorphism

Ind

∼

F[X] ⊗

Here we denote by F[X] the linear span of X i n F[G] (even i f X is not a subgroup) . We shall

construct the analogue for a L i e algebra of the grou p algebra of a group —the universal

enveloping algebra—. First, we construct the tensor algebra T (V ) of a vector space V . For

n ≥ 0 deﬁne T

(V ) to be the n-th tensor power of V , i. e., the vector space

(V ) = V ⊗ . . . ⊗ V



 

n times

Here it is understood that T

(V ) = F and T

(V ) = V .

Deﬁnition 2.7.2. For a vector space V , we deﬁn e the tensor algebra T (V ) by

T (V ) =



n≥0

(V ), (2.10)

with t h e associative algebra structure given by t he condition

⊗ . . . ⊗ v

)(w

⊗ . . . ⊗ w

) = v

⊗ . . . ⊗ v

⊗ w

⊗ . . . ⊗ w

∈ T

m+n

(V ).

Then T (V ) becomes a Z-graded associative algebra with T (V )

= T

(V ), for n ≥ 0, and

T (V )

= 0, for n < 0. Such algebra is characterized by the following u n i versal property:

given any associative a l geb r a A and linear map j : V → A, there exists a unique algebra

map f : T (V ) → A for wh i ch the diagram

T (V )

commutes, where i is the inclusion map of V into T (V ). In the same sense that the tensor

algebra is the ‘universal associative algebra over V ’, the symm etr ic algebra S(V ) is the

universal commutative associative algebra over V . To construct it, let I be the ideal of

T (V ) generated by all the elements v ⊗ w − w ⊗ v, for v, w ∈ V , so that I is the line ar

span of the products a(v ⊗ w − w ⊗ v)b, for a, b ∈ T (V ), v, w ∈ V . Form the algebra

S(V ) = T (V )/I. Since I is spanned by homogeneous elements, then it is clear t hat S(V )

is a Z-graded commutative algebra of the form

S(V ) =



n≥0

(V ), (2.11)

where S

(V ) = S(V )

, called the n-th symmetric power of V , is the image in S(V ) of

(V ). We have S

(V ) = F an d S

(V ) = V . The algebra S(V ) is characterized by a

universal property analogous to th e one above, but for li n e ar maps of V int o commutative

associative algebras. Given a basis {v

}

j∈J

(J a totally ord er ed index set) of V , S(V ) has

a basis con si st i n g of the products V

···V

, for n ≥ 0, j

ℓ

∈ J, j

≤ . . . ≤ j

. The space

(V ) has an obvious basis. If V is G-graded (G an abelian group) then T (V ) and S(V )

acquire unique algebra G-gradings (diﬀerent from (2.10) and (2.11)) extending the grading

of V . We now turn to universal envelop i n g algebras.

Deﬁnition 2.7.3. Given a Lie algebra g, the universal enveloping algebra U(g) is con-

structed as th e quotient associative algebra of T (g) by the ideal generated by the elements

x ⊗ y − y ⊗ x − [x, y], for x, y ∈ g. That is

U(g) = T (g)/I.

Clearly, F embeds in U(g). There is a canonical linear map i : g → U(g) which is an

homomorphism of Lie algebras, and U(g) is characterized by the following universal prop-

erty: Given any associative algebra A and Lie algebra map j : g → A, there is a unique

associative algebra m ap f : U(g) → A maki n g the diagram

U(g)

commute. In particular, every g-modul e is a U(g) - m odule in a natural way and conversely.

If the Lie algebra g is G-graded ( G an abelian group), th en U(g) becomes a G-graded

algebra in a canonical way via the G-grading of T (g). If the Lie algebra g is abelian, then

U(g) is just the symmetric algebra S(g), and in particular, the map i : g → U(g) is an

inclusion and we know a basis of U(g). For a general Lie algebra g, the correspond i ng

result is not trivial, and we need th e following result [116, p.168]:

Theorem 2.7.4 (Poincar´e-Birkhoﬀ- Wi t t) . The canoni cal map i : g → U(g) is injective.

Furthermore, let {x

}

j∈J

(J a totally ordered index set) be a basis of g. Then the universal

enveloping algebra U(g) has a basis consisting of the ordered products x

···x

, for n ≥ 0,

ℓ

∈ J, j

≤ . . . ≤ j

Now we turn to induced Lie algebr a modul es.

Deﬁnition 2 .7 .5. Given a subalgebra h of a Lie algebra g, and a h-module V , the g-module

induced by V i s by deﬁnition the g-module corresponding to the U(g)-mod u l e

Ind

V = U(g) ⊗

U(h)

There is a canonical h-module map i : V → Ind

V , given by v → 1 ⊗ v, and the induced

module is characterized by the following universal property: For any g-modu le W and

h-module map j : V → W , t here is a unique g-module map f : Ind

V → W making the

diagram

Ind

;;

commute. If g, h and V are G-graded (G an abelian group), t h en so is Ind

V , in a canonical

way.

Suppose t hat k and h are subalgebras of g, such that g = k ⊕h as vector sp a ces. Then the

Poincar´e-Birkhoﬀ-Witt theorem (Theorem 2.7.4) implies that the linear map

U(k) ⊗

U(h) → U(g)

deﬁned by x ⊗ y → xy is a linear isomorphism (using a basis of g made u p by bases of k

and h). It follows that the linear m a p

U(k) ⊗

V → U(g) ⊗

U(h)

deﬁned by x ⊗ v → x ⊗ v is a linear isomorphism . The acti on of k on Ind

V carries over

to th e left multiplication action of k on U(k) ⊗ V .

We mention an important special construction . Sup pose that V is a ﬁnite-dim e n si on a l

vector space, with a non -si n g u l ar symmetric bilinear form ·, ·. L et {v

, . . . , v

} be a basi s

of V and let {v

′

, . . . , v

′

} be the corresponding dual basis of V

∗

, deﬁ n ed by

v

, v

′

 = δ

, for i, j = 1, . . . , n.

Thus, th e element



j=1

′

⊗ v

∈ V ⊗ V,

is independent of the choice of basis. In fact, consider the linear isomorph i sm i : V

∗

→ V

from the dual V

∗

to V determi n e d by ·, ·, and the canonical linear isomorphism j :

End V → V

∗

⊗ V given by



i=1

v

, v →



i=1

′

⊗ v) .

Then, ω

= ((i ⊗ 1) ◦ j)(1

), where 1

is denoting the identity in End V . The canonical

image



j=1

′

∈ S

(V )

of ω

in the symmetric square of V is also indepen d e nt of the bas is . In particular, if V

admits an orthonormal basis {e

, . . . , e

} (for instance, if F is algebraically closed) then



⊗ e

and ω



2.8 Aﬃne Lie algebras

Let g be a Lie algebra and ·, · a bilinear form on g —a bilinear map from g × g t o F—.

Then · , · is said to be invariant o r g-invariant i f

(i) ( associativity) [x, y], z = x, [y, z], for x, y, z ∈ g.

Suppose that ·, · is an invariant symmetric bilin ea r form on g. To the pair (g, ·, ·  )

we shall associate two inﬁnite-dimension al graded Lie algebras

g and

g, called the ‘aﬃne

Lie alg eb r as’ .

Let F[t, t

−1

] be the commutative associative algebra of Laurent polynomials in an in deter-

minate t. For a Laurent polynomial

f =



n∈Z

, a

∈ F

the sum being ﬁnite, set f

= a

. Let d be the derivation

d = t∂

(2.12)

on F[t, t

−1

]. Note that (df )

= 0. Consider the vector space

g = g ⊗

F[t, t

−1

] ⊕ Fc,

where Fc is a one-dimensional sp ace. Ther e is an (alternat i n g ) bilinear map [·, ·] :

g×

g →

determined by the condition s

[c,

g] = [

g, c] = 0;

[x ⊗ f, y ⊗ g] = [x, y] ⊗ fg + x, y(df · g)

c; (2.13)

for x, y ∈

g, f, g ∈ F[t, t

−1

], or equivalently,

[c,

g] = [

g, c] = 0;

[x ⊗ t

, y ⊗ t

] = [x, y] ⊗ t

m+n

+ x, ym δ

m+n,0

c; (2.14)

for all x, y ∈

g, m, n ∈ Z. It follows directly from th e symme t ry and invariance of ·, · that

g is a Lie algebra.

Deﬁnition 2.8.1. We will call this Lie algeb r a

g the aﬃne algebra or the untwisted aﬃne

algebra associated with g and ·, ·.

Give the space g ⊗ F[t, t

−1

] the Lie algebra structure by:

[x ⊗ t

, y ⊗ t

] = [x, y] ⊗ t

m+n

, for x, y ∈ g, m, n ∈ Z.

Then, t h er e is an exact sequence of Lie algebras via the canonical maps

0 −→ Fc −→

g −→ g ⊗ F[t, t

−1

] −→ 0,

so that

g is a central extension of g ⊗ F[t, t

−1

For x ∈ g, we shall sometimes write x for the element x ⊗ t

of g ⊗ F[t, t

−1

Suppose that g is not assumed to be a Lie algebra, but only a nonassociative algebra under

[·, ·]. Suppose also that the for m ·, · on g is not assumed symmetric or invariant, but only

bilinear. We can repeat th e construction of th e vector space

g and of t h e nonas sociative

algebra structure [·, ·] given by (2.1 4 ) . In this case,

g is a Lie algebra if and only if g is a

Lie alg eb r a and the form ·, · on g is symmetric and g-i nvariant.

Led d also denote the derivation of

g deter m i n ed by

d(c) = 0,

d(x ⊗ f ) = x ⊗ df, (2.15)

for x ∈ g, f ∈ F[t, t

−1

]. Form the semidirect product Lie algebra

g =

g ⋊ Fd,

called the extended aﬃne algebra associated wi t h g and ·, ·, or just the aﬃne algebra, i f

no con fu si on is possible. We obtain a natural gradation

g =



n∈Z

by considering the eigenspaces

= {x ∈

g : [d, x] = nx}, n ∈ Z,

of ad d. Then, d is th e degree derivation with respect to this grading, and

g =



g ⊗ t

for n = 0

g ⊕ Fc ⊕ Fd for n = 0

where we write g ⊗ t

as g. We also have a gradatio n of

g =



n∈Z

via

∩

When h is a subalgebra of g, we shall consider

h and

h as suba lg eb r as of

g and

g in

the obvious way. We shall also consider the analogue of aﬃnization by ‘twisting’ by an

involution of g.

Deﬁnition 2.8.2. An automorphism θ of a Lie algebra (or another algebraic stru ct ure) is

called an involution if

= 1.

Let θ be an involution of a L i e algebra g which is also an isometry with respect to the form

·, ·, i. e.,

θx, θy = x, y for x, y ∈ g.

For i ∈ Z

, set

(i)

= {x ∈ g : θx = (−1)

x}.

Then,

• g = g

(0)

⊕ g

(1)

;

• [g

(0)

, g

(0)

] ⊆ g

(0)

, [g

(0)

, g

(1)

] ⊆ g

(1)

, [g

(1)

, g

(1)

] ⊆ g

(0)

;

• g

(0)

, g

(1)

 = 0.

Consider th e algebra F[t

1/2

, t

−1/2

] of Laurent polynomials in an indeterminate t

1/2

(whose

square is t), and extend d to a derivation of F[t

1/2

, t

−1/2

] via

d : t

n/2

→

n/2

, n ∈ Z, (2.16)

and form

l = g ⊗

F[t

1/2

, t

−1/2

] ⊕ Fc.

The formulas (2.13) and (2.14) again make l into a Lie algeb r a. Let ν the automorphism

of F[t

1/2

, t

−1/2

] such that

ν : t

1/2

→ −t

1/2

and let θ be the automorphism of l determined by

θ :



c → c

x ⊗ f → θx ⊗ νf

for x ∈ g, f ∈ F[t

1/2

, t

−1/2

]. The formula θ(x⊗f) = θx⊗f will deﬁne another automorph is m

of l.

Deﬁnition 2.8.3. The twisted aﬃne algebra

g[θ] is then the subalgebra

g[θ] = {x ∈ l : θx = x}

of ﬁxed points of θ in l.

We have

g[θ] = g

(0)

⊗ F[t, t

−1

] ⊕ g

(1)

⊗ t

1/2

F[t, t

−1

] ⊕ Fc.

We can again adjoin the derivation d determined by (2.16 ) as in (2.15) and set

g[θ] =

g[θ] ⋊ Fd,

the extended twisted aﬃne algebra associated with g, ·, · and θ. Th e eigenspaces of ad d

make

g[θ] and

g[θ] into

Z-graded Lie algeb r as. Not e that if θ = 1, then

g[θ] degenerates

to th e untwisted aﬃne algebra

Finally, we remark that the process of twisted aﬃnization can be extended to any auto-

morphism of ﬁni t e order of g, which is an isometry with respect to ·, ·.

Chapter 3

The Root space decomposition

Now we will consi d e r a class of Lie algebras (t h e complex semisimple ones), that their

representations can be described, similarly to sl

(C), by a ‘theorem of the highest weight’.

We develop the structures needed to state the theorem of the highest weight. Although

this chapter could be understood simply as a description of the structure of semisimple

Lie algebras, without any mention of representation theory, it is helpful t o have the repre-

sentations in mind. The represe ntation theory, especially in light of sl

(C), motiva t es the

notions of Cartan subalgebras, roots, and the Weyl group. See [179] or [91] a more detailed

treatment of these topics.

3.1 Representations

Deﬁnition 3.1.1. Let G be a matrix Lie group. Then, a (ﬁnite-dimension al ) complex

representation of G i s a Lie group h om om o r p h i sm Π : G → GL

other words, a Lie group homomorphi sm Π : G → GL(V ), where V is a ﬁnite-dimensi on a l

complex vector space (wit h dim V ≥ 1). A ﬁnite-dimensional real representation of G

is a Lie group homomorphism Π of G into GL

(R) or into GL(V ), where V is a ﬁnite-

dimensional real vector space.

Deﬁnition 3.1.2. If g is a real or complex Lie algebra, th e n a ﬁn i t e-dimensional complex

representation of g is a Lie al gebra homom or p h i sm π : g → gl

V i s a ﬁnite-dimensional complex vector space. We can deﬁne a real representation of g in

a similar way. If Π or π is a on e- t o- one homomorphism, then the representation is called

faithful.

One should think of a representation as a linear action of a group or Lie algebra on a vector

space (since to every g ∈ G, there is associated an operator Π(g), which acts on the vec-

tor space V ). If g is a real Lie algebra, we will consi d er mainly complex representations of g.

Deﬁnition 3.1.3. Let Π be a ﬁnite-dimensional real or com p l ex representation of a m a tr i x

Lie group G, acting on a space V . A subspace W of V is called invariant if Π(A)w ∈ W ,

for all w ∈ W and all A ∈ G. An invariant subspace W is call ed nontrivial if W = {0}

and W = V . A r ep r esentation with no nontrivial invariant subspaces is called irreducible.

The terms i nvariant, nontrivial, and irreducible are deﬁned analogously for representations

of Lie algebras.

Deﬁnition 3.1.4. Let G be a matrix Lie group, let Π be a representation of G acting on

the space V , and let Σ be a representation of G acting on the sp ace W . A linear map

ϕ : V → W is called an intertwining map of representations if



Π(A)v



= Σ(A)ϕ( v),

for all A ∈ G and all v ∈ V . The analogous proper ty deﬁnes intertwining maps of repre-

sentations of a Lie algebra. If ϕ is an intertwining map of representations and, in addition,

ϕ is invertible, then ϕ is said to be an equivalence of representati on s . If there exists an

isomorphism between V and W , then the representations are said to be equivalent.

If G is a matrix Lie group with Lie algebra g and Π is a (ﬁnite-dimension a l real or complex)

representation of G, acting on the space V , then there is a un i qu e representation π of g

acting on the same space, such that

Π(e

) = e

π(x)

, for all x ∈ g.

This representation π can be comp u t ed as

π(x) =

Π(e

)



t=0

and satisﬁes π( AxA

−1

) = Π(A)Π(x)Π(A)

−1

, for all x ∈ g and all A ∈ G. We state the

following two propo si ti o n s. For a proof of them, see [91, p.93].

Proposition 3.1.5. 1. Let G be a connected matrix Lie group with Lie algebra g. Let

Π be a representation of G and π the associated representation of g. Then, Π is

irreducible if an d only if π is irreducible.

2. Let G be a connected matrix Lie group, let Π

and Π

be representations of G, and

let π

and π

be the associated Lie algebra representations. Then, π

and π

are

equivalent if and only Π

and Π

are equivalent.

Proposition 3 .1 .6. Let g be a real Lie algebra and g

its complexiﬁcation. Then, every

ﬁnite-dimensional complex representation π of g has a unique extension to a complex-linear

representation of g

, also denoted π and given by

π(x + iy) = π(x) + iπ(y), for all x, y ∈ g.

Furthermore, π is irreducible as a representation of g

if and only if it π irreducible as a

representation of g.

We give some examples of representations:

Example 3.1.7. (The st andard representation)

A matrix Lie group G is, by deﬁni ti o n , a su bset of some GL

(C). The inclusion map of G

into GL

of G. If G happens to be contained in GL

(R) or GL

(C), then we can think of the stan-

dard representation as a real representation if we prefer. Thus, for example, the standard

representation o f SO

and the

standard representation of SU(2) is the one in which SU(2) acts on C

in the usual way.

If G is a subgroup of GL

(R) or GL

(C), then its Lie algebra g will be a subalgebra of

(R) or gl

(C). The inclusion of g into gl

(R) or gl

the standard representation.

Example 3.1.8. (The t r i vi al representation)

Consider the one-d im e n si on a l complex vector space C. Given any matrix Lie group G, we

can deﬁne the trivial representation of G, Π : G → GL

(C), by the formula

Π(A) = I, for all A ∈ G.

Of course, this is an irreducib l e representatio n , since C has no nontrivial subspaces, and

thus no nontrivial invariant subspaces. If g is a Lie algebra, we can also deﬁne the trivial

representation of g, π : g → gl

(C), by

π(x) = 0, for all x ∈ g.

This is an irreducible representation.

Example 3.1.9 (The adjoint representation). Let G be a ma tr i x Lie group with Lie algebra

g. We deﬁne the adjoint mapping Ad : G → GL(g) by the formula

(X) = AXA

−1

Since Ad is a Lie group homomorphism into a group of invertible operators, we see that,

in fact, Ad is a repr ese ntation of G, actin g on the space g. We call Ad the adjoint repre-

sentation of G. The adjoint representation is a real representation of G (if g is a complex

subspace of C

n×n

, th en we can think of t h e adjoint representation as a complex r ep r es en -

tation). Similarly, i f g is a Lie algebra, we have the adjoint map ad : g → gl(g) deﬁned by

the formula (see Chapter 2)

(y) = [x, y].

We know that ad is a Lie algebra homomorphism and is, therefore, a representation of g,

called the adjoint representation. In the case that g is the Lie algebra of some matrix Lie

group G, Ad and ad are related by e

= Ad

Now we will discuss a cla ssi cal example, n am el y the irreducible complex representations of

the Lie algebr a su(2). This computation is im portant for several reasons. In the ﬁrst place,

su(2)

∼

so(3) and the representations of so(3) are of physical signiﬁcance, particularly

in quantum mechanics [91]. In the second place, the representation theory of su(2) is an

illuminating example of how one uses relations to determine the representations of a Lie

algebra. Also, in determining the representations of semisimple Lie a lg eb r as (Section 3.2),

it usually uses the representation of su(2).

Example 3.1.10. (The Ir r educible Repr esentations of sl

(C))

Every ﬁnite-dimensi onal complex r epresentation π of su(2) extends to a comp le x- l inear

representation (also called π) of the com p l exi ﬁ ca t io n of su(2), namely sl

(C). The extensio n

of π to sl

use the following basis for sl

(C):

h =



1 0

0 −1



, x =



0 1

0 0



, y =



0 0

1 0



;

which have the commutation relations

[h, x] = 2 x,

[h, y] = −2y,

[x, y] = h.

If V is a (ﬁnite-dimensional complex) vector space and A, B and C are operators on V

satisfying

[A, B] = 2B,

[A, C] = −2C,

[B, C] = A,

then because of the skew symmetry and bilineari ty of brackets , the linear map π : sl

gl(V ) satisfying

π(h) = A, π(x) = B, π(y) = C,

will be a representation of sl

(C).

We state an important result we will use later. For a proof see [179 , p.268].

Theorem 3.1.11. Suppose π is any ﬁnite-dimensional, complex-linear representation of

1. Every eigenvalue of π(h) must be an integer.

2. If v is a nonzero element of V such that π(x)v = 0 and π(h)v = λv, then λ is

a non-negative integer. Furthermore, the vectors v, π(y)v, . . . , π(y)

v are linearly

independent and their span is an irreducible invariant subspace of dimension λ + 1.

Analogously to the previous example, we can think of study the representations of su(3) by

studying the representations of its complexiﬁcation sl

(C). We will discuss this particu l ar

case in Section 3.9. In general, studying the irreducible representations of su(n) is equiva-

lent to st udying the ir r educible (complex-linear) representations o f sl

(C). Passing to the

complexiﬁed Lie algebra m akes our comput a t io n s easier, and we can ﬁnd a nice basis for

3.2 Complete Reducibility and Semisi mpl e algeb ra s

Deﬁnition 3.2.1. A ﬁnite-dimensional representation of a group or Lie algebr a is said to

be completely reducible if it is isomorphic to a direct sum of a ﬁnite number of irreducible

representations. A group or Lie algebra is said to have the complete reducibility property if

every ﬁn it e- d i m ensional repres entation of it is completely reducible.

The complete reducibility property is a very special on e that most groups and Lie algebras

do n ot have. If a group or Lie algebra does have the complete reducibili ty property, then

the study of its representations redu ces to the study of its irreducible representations, which

simpliﬁes the an al ys is considera b l y. The following results are useful [91, p.1 19]

Proposition 3.2.2. If V is a completely reducible representation of a group or Lie algebra,

then the following properties hold:

1. Every invariant subspace of V is completely reducible.

2. Given any invariant subspace U of V , there is another invariant subspace

U such that

V = U ⊕

Proposition 3.2.3. Let G be a matrix Lie group. Let Π be a ﬁnite-dimensional unitary

representation of G, act ing on a ﬁnite-dimensional real or complex Hilbert space V . Then,

Π is completely reducible.

If Π is a repr esentation of a ﬁnite gr ou p G, acting on a space V , we can choose an ar b i t r ar y

inner product ·, · on V . Then, we can deﬁn e a new inner product ·, ·

on V by

v

, v





g∈G

Π(g)v

, Π(g)v

. (3.1)

Furthermore, if h ∈ G, then

Π(h)v

, Π(h)v





g∈G

Π(g)Π(h)v

, Π(g)Π(h)v





g∈G

Π(gh)v

, Π(gh)v



However, as g rang es over G, so d oes gh. Thus, in fact, Π(h)v

, Π(h)v



= v

, v



;

that is , Π is a uni t ar y representation with respect to the inner product ·, ·. Thus , Π is

isomorphic to a direct sum of irreducibles, by Proposition 3.2.3 and we have

Proposition 3.2.4. Every ﬁnite group has the complete reducibility property.

There is a variant of the above argument which can be used to prove the following result

[116]:

Proposition 3.2.5. If G is a compact matrix Lie group, G has the complete reducibility

property.

The argument below is sometimes called ‘Weyl’s Unitarian trick’. Its proof requires the

notion of Haar measure (see, for example, [116]). A left H aar measure on a matrix Lie group

G is a nonzero measure µ on the Borel σ-alg eb r a in G with the fol lowing two properties:

• It is locall y ﬁnite (i. e., every point i n G has a neighborhood with ﬁnite measure),

• It is left-transl at i on invariant: µ(gE) = µ(E), for al l g ∈ G and all Bor el sets E ⊆ G.

It is a fact , wh i ch we cannot prove here, that every matrix Lie group has a left Haar measure

and t h at this measure is unique up to multiplication by a constant. One can analogously

deﬁne right Haar measure, and a similar theorem holds for it. Left Haar measure and right

Haar measure may or may not coincide (a group for which they do is cal l ed unimodular).

Now, the key fact for our purpos e is t h a t le ft Ha ar measure is ﬁnite if and only if the group

G is compact. Suppose, then, that Π is a ﬁnite-dimensional representation of a compact

group G acting on a space V . Let  · , · be an arbitrary inner product on V and d eﬁne a

new in n er produ ct ·, ·

on V (analogous to (3.1)) by

v

, v





Π(g)v

, Π(g)v

dµ(g), (3.2)

where µ is a left Haar measure. Again, it is possible to check that ·, ·

is an inn er product.

Furthermore, if h ∈ G, then by the left-invariance of µ,

Π(h)v

, Π(h)v





Π(g)Π(h)v

, Π(g)Π(h)v

dµ(g)



Π(gh)v

, Π(gh)v

dµ(g) = v

, v



So, Π is a unitary representation with respect to  · , ·

and t hus completely reducible (note

that t h e integral in (3.2) is convergent beca u se µ is ﬁnite).

Recall from Chapter 2 t h a t if g is a complex L i e algebra, then a n ideal in g is a co m p l ex

subalgebra h of g with the property that for all x ∈ g and all h ∈ h, we have [x, h] ∈ h.

Recall also that a comp l ex Lie algebra g is called indecomposable if the only ideal s in g

are g and (0) . A complex Lie algebra g is call ed simple if g is indecomposable and dim g ≥ 2.

There is an analogy between ﬁnite-dimensional Lie algebras an d ﬁn i t e group s. Subalgebras

in the Lie algebra setting are the analogs of subgroups in the ﬁnite group setting , and ideals

in the Lie algebra setting are the analogs of normal subgroups in the ﬁn i t e group set t i n g. In

this ana lo gy, th e one-dimension a l Lie algebras (which are precisely the Lie algebras having

no nontrivial subal g eb r as) are the analogs of the c ycl i c groups of prime order (which are

precisely the gr ou ps having no nontrivial subgroups). However, there is a discrepancy in

terminology: cyclic groups of prime order a r e called simple but one-dimensional Lie algebras

are not called simple. This terminological convention is importa nt to bear in mind in the

following deﬁnition.

Deﬁnition 3.2.6. A complex Lie algebra is calle d reductive if i t is isomorphic to a direct

sum of indecomposable Lie algeb r as . A compl ex Lie algebra is called semisimple if it

isomorphic to a direct sum of simple Lie algeb r a s.

Note that a reductive Lie algebra is a direct sum of indecomposable algebra s, which are

either si m p l e or one-dimensional commutative. Thus, a reductive Li e algebra is one that

decomposes as a direct sum of a semisimple algebra (coming from the sim p l e terms in the

direct sum) and a commutative al ge b r a (coming from the one-dimensional terms in the

direct sum). We will assume that the complex semi si m p l e Lie algebras we study are given

to us as subalgebras of some gl

(C), since by Ado’s Theorem 2.3.7 every ﬁnite-dimensional

Lie algebra has a faithful ﬁnite-dimensional representation. In fact, for semisimple Lie

algebras, the adjoi nt representation is always faithful, as is shown in [91, p.158]

Proposition 3.2. 7. A complex Lie algebra g is reductive precisely if the adjoint represen-

tation is completely reducible.

In fact, the complexiﬁcation of the Lie algebra of a connected compact matrix Lie gr ou p

is reductive. This follows from the above proposition and the p r operty that connected

compact groups have the complet e redu cibility property (P r oposition 3.2.5). Note that

the Lie algebra of a compact Lie group may be only reductive and not semisimple. For

example, the Lie algeb r a of S

is on e dimensional and, thus, not semisimple. We have the

following characterization result of semisimp l e Lie algebras (see [179, p.348])

Theorem 3.2.8. A complex Lie algebra is semisimple if, and only if, it is isomorphic to

the complexiﬁcation of the Lie algebra of a simply-connected compact matrix Lie group.

We have already seen that if g is the complexiﬁcation of the Lie algebra of a compact

simply-connected group K, then g is reductive, even if K is not simply connected. Thus

g = g

⊕g

, wit h g

semisimple and g

commutative. It can be shown that the Lie algebra k

of K decomposes as k = k

⊕k

, where g

= k

+ ik

and g

= k

+ ik

. Then, K dec om poses

as K

×K

, where K

and K

are simply connected and where K

is commutative. However,

a simply-connected commutative Lie g r ou p is isomorphic to R

, which is noncompact for

n ≥ 1. Thus, the compactness of K means that k

= {0}, in which case g

= {0} and

g = g

is semisi m ple.

For the other direction, given a complex sem i si m p l e Lie algebra, we must ﬁnd the correct

real form whose correspondi n g simply-con n ect ed group is compact (c. f. [179]).

Deﬁnition 3.2.9. If g is a complex semisimple Lie algebra, then a compact real form of

g is a real subalgeb r a k of g with the property that every x ∈ g can be written uni quely

as x = x

+ ix

, with x

and x

in k and such that there is a compact simply-connected

matrix Lie group K

′

such tha t the Lie algebra k

′

of K

′

is isomo r p h i c to k.

We have the following important fact [91, p . 159 ].

Proposition 3.2.10. Let g be a complex semisimple Lie algebra. If g is a subalgebra of

(C)

whose Lie algebra is k is compact.

In particular, every complex semisimple Lie algebra has the complete reducibility property.

This last statement holds because the representations of g are in one-to-one correspondence

with the representations of K, and compact groups h ave the complete reducib il i ty prop-

erty (Proposi t i on 3.2.5). Actually, only the semisimple ones have the complete reducibility

property, and thus, complete redu c ibility is sometimes taken a s the deﬁnition of sem i si m -

plicity for Lie algebras. For an algebraic proof of complete red u ci b i l i ty of semisimp l e Lie

algebras, see [100]). Up to now, we have considered only complex semisimple Lie algebras,

since these are the ones whose representations we will co n si d er . Nevertheless, we can d eﬁ ne

the terms ideal, indecom posable, simple, r ed uctive, and semisimple for real Lie algebras in

precisely the same way as for the comp l ex case.

Let us consider some examples of Lie algebr as that are red u c t ive or semisimple. The

following table lists some of the co m p l ex Lie algebras that we have encountered alr eady that

are either reductive or semisimple (see [22]). Her e, ‘reductive’ means actually ‘reductive

but n o t semisimple’ .

Group Reductive/Semisimple

Table 3.1: Semi si m p l e proper t i es of some classical complex Lie al geb r a s.

The other Lie algebras we have examined, such as the Lie algebras of th e Heisenberg group,

are nei t h er reductive nor semisimple.

Group Reductive/Semisimple

su(n) (n ≥ 2) semisimple

so(n) ( n ≥ 3) semisimple

so(2) reductive

sp(n) ( n ≥ 1) semisimple

(R) (n ≥ 1) semisimple

(R) (n ≥ 2) semisimple

(R) (n ≥ 1) reductive

Table 3.2: Sem i si m p l e propert i es of some classical real Lie algebr a s.

3.3 Cartan subalgebras

Deﬁnition 3.3.1. If g is a comp l ex semisimple Lie alge b r a, then a Cartan subalgebra of g

is a complex subspace h of g with the following properties:

(i) For all h

and h

in h, [h

, h

] = 0.

(ii) For all x ∈ g, if [h, x] = 0 for all h ∈ h, then x is in h.

(iii) For all h ∈ h, ad h is diagonalizable.

Condition (i) says that h is a commutative subalgebra of g. Condition ( ii ) says that h is

a maximal commutative subalgebra, i. e., not contained in any larger commutative sub-

algebra. Condition (iii) says that each ad h (h ∈ h) is diagonalizable . Since the h’s in h

commute, the ad h’s also commute, and thus they are simultaneously diagonalizable. (It is

a standard result in linear algebra that any commuting fami l y of diagonalizable matrices is

simultaneously di ago n al i zable; see [97]) . Of cour se , the deﬁnition of a Cartan subalgebra

makes sense in any Lie algebra, semisimple or not. However, if g is not semisimple, then g

may not have any Cartan su b a l geb r a s. Even i n the semisimple case we must pr ove tha t a

Cartan subalgebra exists (see [91, p.163]).

Proposition 3.3.2. Let g be a complex semisimple Lie algebra, let k be a compact real form

of g, and let t be any maximal commutative subalgebra of k. Deﬁne h ⊆ g to be h = t + it.

Then, h is a Cartan subalgebra of g.

Note that k (or any other Lie algebra) contains a maximal commutative subalgebra. After

all, let t

be any one-dimensional subspace of k. Then, t

is a commutative subal geb r a of k.

If t

is maximal, then we are done; if not, then we choose some commutative subalgebra t

properly containing t

. Th en , i f t

is maxim al , we are done, and if not, we choose a com-

mutative subalg eb r a t

properly containing t

. Since k is ﬁni t e dimensional, this process

cannot go on forever and we will eventually get a maximal commutative subalgebra.

It is possible to prove that every Cartan subalgebra of g arises as in Proposition 3.3.2 (for

some compact real form k and some maximal co m mutative subalgebra t of k) and also th at

Cartan subalgebras are unique up to conjugation. In particular, all Cartan subalgebra s of

a given complex semisimple Lie al gebra have the same di m en si o n . In light of this result,

the foll owing deﬁniti on makes sense.

Deﬁnition 3.3.3. If g is a complex semisimple Lie algebra, then the rank of g is t he

dimension of any Cartan subalgebra.

3.4 Roots and Root Spaces

From now on we assume that we have chosen a compact real form k of g and a maximal

commutative subalgebra t of k, and we consider the Cartan su b a lg eb r a h = t + it. We

assume also that we have chosen an inner product on g t h at is invariant under the adjoint

action of K and that takes real values on k.

Deﬁnition 3.4.1. A root of g (relative to the Cartan subalgebra h) is a nonzero linear

functional α on h such that there exists a nonzero element x of g with

[h, x] = α(h)x,

for all h ∈ h.

The set of all roots is denoted by Φ. The cond it i o n on x says that x is an eigenvector for

each ad h, with eigenvalue α(h). Note that i f x is actually an eigenvector for each ad h with

h ∈ h, then the eigenvalues must depend lin ea rl y on h. That is why we insist that α be a

linear functional on h. So, a root is just a (nonzero) collection of simultaneous eigenvalues

for the ad h’s. Note that any elem ent of h is a simultaneous eigenvector for all the a d h’s,

with all eigenvalues equal to zero, but we only call α a r oot if α i s nonzero. Of cours e, for

any root α, some of the α(h)’s may be equal to zero; we just require that not all of them

be zero. Note tha t th e set of li n ear functionals on h that are imaginary on t forms a real

vector space whose real dimension equals the complex dimension of h. If t

∗

denotes the

space of real-valued linear functionals on t, then the r oots ar e contained in it

∗

⊆ h

∗

Deﬁnition 3.4.2. If α is a root of the Lie algebra g (relati vely to the subalgebra h), then

the root space g

is the space of all x ∈ g for which [h, x] = α(h)x, for all h ∈ h. An

element of g

is called a root vector (for the root α).

More generally, i f α is any element of h

∗

, we deﬁne g

to be the space of all x ∈ g for which

[h, x] = α(h)x, for all h ∈ h (but we do not call g

a root space unless α is actually a

root). Taking α = 0, we see that g

is the set of all elements of g th at commute with every

element of h. Since h is a maximal commutative subalgebra, we conclude t h a t g

= h. If

α is not zero and not a root, then g

= {0}. Now, since h is commutative, the operators

ad h, h ∈ h, all commute. Furthermore, by th e deﬁnition of Cartan subalgebra, each ad h,

h ∈ h, is diagonalizable. It follows that the ad h’s, are simultaneously diagonalizable. As a

result, g can be decomposed as the direct sum of h and the root spaces g

g = h ⊕



α∈Φ

This means that every el em ent of g can be written uniquely as a sum of an element o f h

and on e element from each root space g

We resume some elementary properti es of roots. You usually can ﬁnd a proof of theses

properties in almost books abo u t r epresentation theory. See for example [91] or [59] for the

proofs.

Proposition 3.4.3. (i) For any α and β in h

∗

, [g

, g

] ⊆ g

α+β

(ii) If α ∈ h

∗

is a root, then so i s −α.

(iii) If α is a root, then the only multiples of α that are roots are α and −α.

(iv) The roots span h

∗

(v) If α is a root, then the root space g

is one dimensional.

(vi) For each root α, we can ﬁnd nonzero elements x

∈ g

, y

∈ g

−α

, and h

∈ h such

that

, x

] = 2x

, [h

, y

] = −2y

, [x

, y

] = h

The element h

is unique, i. e., in dependent of the choice of x

and y

Last point of the p r o position above tells us that x

, y

, and h

span a subalgebra of g

isomorphic to sl

(C). The elements h

of h are called the co-roots. Their properti es are

closely related to the properties of the roots themselves.

Given any linear functional α ∈ h

∗

(not n eces sar i l y a root), there exists a unique element

∈ h such that

α(h) =  h

, h,

for all h ∈ h, where we take the inner product to be linear in the second factor. The map

α → h

is a one-to- o n e and onto correspondence between h

∗

and h. However, this correspon-

dence is not linear but ra ther conjugate-linear, since the inner product is conjugate-linear

in the ﬁrst factor (where h

is). It is convenient to permanently identify each root α ∈ h

∗

with the corresponding el em ent h

∈ h. Having done t his, we then omit the h

notation

and d en ot e that element of h simp ly as α.

The reader can ﬁnd some more proper t i es and relations of roots and co-roots in [91]. We

only mention that if α ∈ h is a root in the sense o f last paragrap h and h

is the corresponding

co-root, then α and h

are rela t ed by the formulas

2α

α, α

, α =

h

, h



. (3.3)

The real content of this proposition is that once we use the inner product to identify h

∗

with h (so that the root s and co-roots now live in t h e same space), α and h

are multiples

of one another. Once this is known, the normalizat i on is determined by the conditio n

that α, h

 = 2, which reﬂects that [h

, x

] = 2x

. Observe t h a t both formulas ( 3 .3 ) are

consistent with the relation α, h

 = 2. We conclude with the following [91, p.173]

Theorem 3.4.4. For all roots α, β ∈ h (in the notation above), the quantities

α, β

α, α

and 2

h

, h



h

, h



are integers and, furthermore,

α, β

α, α

= 2

h

, h



h

, h



3.5 The Weyl Group

We use here the comp act - gr oup approach to deﬁning the Weyl group, as opposed to th e

Lie algebra approach. The compact-group approach makes certain aspects of the Weyl

group more transparent. Never t h el ess , th e two approaches are equivalent. We continue

with the setting o f the previous section. Thus, g is a complex semisimple Lie algebra given

to us as a subalgebra of some gl

(C). We have chosen a compact real form k of g and

we let K be the compact subgroup of GL

maximal commutative subalgebra t of k, and we work with the associated Cartan subal gebra

h = t + it. We have chosen an inner product on g that is invariant un der the adjoint action

of K and that takes real values on k. Consid er the following two subgroups of K:

Z(t) = {A ∈ K : ad A( h) = h, ∀h ∈ t},

N(t) = {A ∈ K : ad A(h) ⊆ t, ∀h ∈ t}.

Clearly, Z(t) is a normal su bgroup of N(t). If T is the con n ect ed Lie subgroup of K with

Lie algebra t, then T ⊆ Z( t) , since T is generated by elements of the form e

with h ∈ t.

It tur ns out that, in fact, Z(t) = T . See [22].

Deﬁnition 3.5.1. The Weyl group for g is t h e quotient group W = N(t)/Z(t).

We can deﬁne an action of W o n t as follows. For each element w ∈ W , choose an element

A of the corresponding equivalence class in N(t). Then for h ∈ t we deﬁne the act io n of w

on h by

w · h = ad A(h).

In fact, this action is well deﬁned (i. e., independent of the choice of A in a given equivalence

class). Since h = t+it, each linear transformation of t extends uniquely to a complex-linear

transformation of h. Thus, we also think o f W as acting on h. If w is an element of the

Weyl group, then we write w · h for t he action of w on an element h of h. It can be seen

that W is isomorphic to the group of linear transformations of h that can be expr ess ed as

ad A for some A ∈ N( t). The following states basic properties of t h e Weyl group [91, p.174]

Proposition 3.5.2. 1. The inn er product ·, · on h is invariant under the action of W .

2. The set Φ ⊆ h of roots is invariant under the action of W .

3. The set of co-roots is invariant under the action of W , and w · h

= h

w·α

, for all

w ∈ W , α ∈ Φ.

4. The W eyl group is a ﬁnite group.

We state some an important property, leading to a ‘dual ’ nature of roots [91, p.178]:

Proposition 3.5.3. For each root α, there exists an element w

of W such that

· α = −α

and such that

· h = h,

for all h ∈ h with α, h = 0.

Note that since h

is a multiple of α, saying w

· α = −α is equivalent to saying that

·h

= −h

. The linear operator corresponding to the action of w

on h is ‘the reﬂection

about the hyperplane perpendicular to α’. This means that w

acts as the identity on the

hyperplane (of codimen si on one) perpendicular to α and as minus the identity on the span

of α. We can work out a formula for w

as follows. Any vector β can be decomposed

uniquely as a multiple of α plus a vector orthogonal to α. This decomposition is given

explicitly by

β =

α, β

α, α

α +



β −

α, β

α, α



, (3.4)

where the second term is indeed orthogona l to α. Now, t o o b t ai n w

·β, we should change

the sign of the part of β parallel to α and leave alone the part of β that is orthogonal to α.

This means that we change the sign of the ﬁrst ter m on the right-hand side of (3.4), giving

· β = β − 2

α, β

α, α

α. (3.5)

We now have another way of thinking about the quantity 2

α,β

α,α

in Theorem 3.4.4: it is the

coeﬃcient of α in the expression for w

·β. So, we can r e- ex p r ess Theorem 3.4.4 as follows.

Corollary 3.5.4. If α and β are roots, then β − w

· β is an integer multiple of α.

Finally, we state a useful characterization of the Weyl group [116, p.208]:

Theorem 3.5.5. The W eyl group W is gen erated by the elements w

as α ranges over all

roots.

That is to say, th e smallest subgroup of W that contains all of the w

’s is W itself. This is

somewhat involved to prove and we will not do so here; see [22] or [116]. In the Lie algebra

approach to the Weyl group, the Weyl group is deﬁned as the set of line ar transform at i ons

of h generated by the reﬂections w

. Theorem 3.5.5 shows that the Lie algebra deﬁnition

of the Weyl group gives the same group as the compact-group approach.

3.6 Root Systems

In the previous section we have esta b l is h ed several properties of roots. For example, we

know that the roots are imaginary on t, which, after transferring the roots from h

∗

to h,

means that t h e roots live in it ⊆ h. The inner product ·, · was constructed to t ake real

values on k, and hence on t. The inner product th en also takes real val u e s on it, since

ix, iy = (−i)iw, y = x, y. So, the roots live in the real inner-product space E = it.

From Proposition 3.4.3 we know that the roots span it and that if α is a root, then −α

is the only other multiple of α also root. Furthermore, Theorem 3. 4. 4 tell us that for any

roots α and β, the number 2

α,β

α,α

is an integer. Finally, we have established that the roots

are invariant under th e action of the Weyl group, and Theorem 3. 5. 5 tells us that the

Weyl group contains the reﬂ ect i on about the hyperp l an e orthogonal to each root α. We

summarize these results in the following theorem.

Theorem 3.6.1. The roots of g form a ﬁnite set of nonzero elements of a real inner-product

space E and have the following properties:

1. The roots span E.

2. If α is a root, then −α is a root and the only multiples of α that are roots are α and

−α.

3. If α is a root, let w

denote the linear transformation of E given by w

·β = β−2

α,β

α,α

α.

Then, for all roots α and β, w

· β is also a root.

4. If α and β are roots, then the quantity 2

α,β

α,α

is an integer.

Deﬁnition 3.6.2. Any collection R of vectors in a ﬁnite-dim e n si on a l real inner-product

space h aving these properties of Theorem 3.6.1 is call ed a root system.

The Weyl grou p for a root system R is the group of linear transformations of E generated

by the w

’s. Note that it em 4 is equiva le nt to saying that β − w

· β must be an integer

multiple of α for all roots α and β. We have also established certain important properties

of the root spaces that are not properti es of the roots th em se lves, namely that each root

space g

is on e dimensi on a l and that out of g

, g

−α

, and [g

, g

−α

], we can form a subal geb r a

isomorphic to sl

(C). Finally, we claim that the co-roots h

themselves form a root system.

Theorem 3.4.4 tells us that the co-roots satisfy property 4 and Proposition 3.5.2 tells us

that the set of co-roots is invari ant under the Weyl group a n d hence, in particular, under

the reﬂe ct io n s w

. However, note that since h

is a multiple of α, the reﬂ ect i o n generated

by h

is the same a s the reﬂecti on gen er at ed by α. Thus, the set of co-roots satisﬁes prop-

erty 3. Properties 1 and 2 for the co-roots follow from the corresponding properties for th e

roots, since ea ch h

is a multiple of α. The set of co-roots is called the dual root system

to t h e set of roots. See Section 3.9 for more information on root systems, including some

pictures.

In the next section, we will present the irreducible representations of g in terms of a ‘highest

weight’. What we need is simply some consistent not i on o f higher and lower that will allow

us to divide the root vectors x

into ‘raising operato rs ’ and ‘lowering oper a t or s’ . This

should be done in such a way that the commut at o r of two raising oper at o rs is, again, a

raising operator and not a lowering operator. This means that we want to divid e the roots

into two groups, one of wh i ch will be called ‘positive’ and the other ‘negative’. This should

be done is s u ch a way that if th e sum of positive root s is again a root, that root shoul d

be positive. There is no un i que way to make the division into positive and negative; any

consistent div i si on will do. The uni q u en e ss theorems of th e next section show that it does

not reall y matter which choice we make. The following deﬁnition and theorem shows that

it is possibl e to make a good choice.

Deﬁnition 3.6.3. Suppose that E is a ﬁnite-dimensional real inner-product space and that

R ⊆ E is a root system. Then, a base for R is a subset ∆ = {α

, . . . , α

} of R such that ∆

forms a basis for E as a vector space and such that for each α ∈ R, we have

α = n

+ . . . + n

where n

∈ Z and either all n

≥ 0 or all n

≤ 0.

Once a base ∆ has been chosen, the α’s for which n

≥ 0, ∀j, are called t h e positive roots

(with respect to the given choice of ∆) and t h e α’s with n

≤ 0, ∀j, are cal l ed the negative

roots. The elements of ∆ are called the positive simple roots. We will d en o t e R

the set

of all positive root s, and R

−

the set of all negative roots, so then R is the disjoi nt union

R = R

∪ R

−

. To be a ba se (in the sense of root systems), ∆ ⊆ R must in p ar t i cu l a r be

a basis for E in the vector sp ace sense. In addition, the exp a n si on of any α ∈ R in terms

of the elements of ∆ must have integer coeﬃcients and all of the nonzero coeﬃcients must

be of the same sign.

3.7 Integral and Dominant integral elements

Deﬁnition 3.7.1. An element ω of h is called an in tegral ele m ent i f ω, h

 is an integer,

for each root α.

As explained in next, the integral elements are preci sel y the elements of h that arise as

weights of ﬁnite-dimensional rep r ese ntations of g. In fact, we can prove that the set of

integral elements is invariant under the action of the Weyl group. Checking that (ω, h

)

is an i nteger for every root α is a rather tiresome process. Fortunately, it suﬃces to check

just for the positive simple roots: if ω is an element of h for which ω, h

 is an integer

for all positive simple roots α, then ω, h

 is an integer for al l roots α, and thus ω is an

integral element.

Recalling the expression ( 3. 3) for h

in terms of α, we may restate a characterizat i on of

integral elements as follows: an element ω ∈ h is integral if and on l y if

ω, α

α, α

is an integer for each positive simple root α. In particular, every root is an integral element.

Recall now from elementary linear algebra that if ω and α are any two elements of an inner-

product space, then th e orthogonal projection of ω onto α is given by

ω,α

α,α

α. Thus, we

may reformulate the notion of an integral element yet again as: ω is integral if and only

if the orthogonal projection of ω onto each positive simple root α is an integer or half-

integer multiple of α. Th i s characterization of the integral elements will help us visualize

graphically what the set of integral elements looks like in example (see Section 3.9).

Deﬁnition 3.7.2. An element ω of h is called a dominant integral element if ω, h

 is a

non-negative integer for each positive simple root α.

Equivalently, ω is a dom i n ant integral element if 2

ω,α

α,α

is a non-negative integer for each

positive simple root α. If ω is dominant integral, then ω, h

 will au t om a t ica l ly be a

non-negative integer for each positive root α, not just th e positive simple ones.

Deﬁnition 3.7.3. Suppose π is a ﬁ nite-dimensional representation of g on a vector space

V . Then, ω ∈ h is called a weight for π if there exists a nonzero vector v ∈ V such that

π(h)v = ω, hv,

for all h ∈ h. A nonzero vector v satisfying condition above is called a weight vector for the

weight ω, and the set of all vectors (zero or nonzero) satisfying this conditio n is called the

weight space with weight ω. The di m en s io n of the weight space is call ed the multiplicity of

the weight.

To understan d this deﬁnition, suppose that v ∈ V is a simultaneous eigenvector for each

π(h), h ∈ h. This means that for each h ∈ h, there is a number λ

such that π(h)v = λ

Since the representation π(h) is li n ea r in h, the λ

’s must depend linearly on h as well;

that is, the map h → λ

is a linear fun ct i onal on h. Then, there is a u n i qu e element ω of

h su ch that λ

= ω, h . Thus, a weight vector is nothing but a simultaneous eigenvector

for all the λ

’s and the vector ω is simp l y a conven ient way of encod ing the eigenva l u es.

Note that the roots (in the dual notation) are pre cis el y the nonzero weights of the adjoint

representation of g. It can be shown that two equivalent representations h ave th e same

weights and multiplici t i es. It is tru e, although by no means obvious, that ever y integral

element actually arises as a weight of some ﬁnite-dimension al representation of g, see [91].

Then, for any ﬁnite-dimensional representation π of g, the weights of π and t heir multiplicity

are inva r ia nt under the action of the Weyl group.

Deﬁnition 3.7.4. Let ω

and ω

be two elements of h. We say that ω

is higher than ω

(or, equival ently, ω

is lower than ω

) if there exist non-negative real numbers a

, . . . , a

such tha t

− ω

= a

+ . . . + a

where ∆ = {α

, . . . , α

} is the set of positive simple root s . This relationship is oft en written

as ω

 ω

or ω

 ω

If π is a representation of g, th en a weight ω

for π is said to be a highest weight if for all

weights ω of π, ω  ω

. We now state an important result of this chapter [91, p.197]

Theorem 3.7.5 (Th eo re m of the Highest Weight). We have:

1. Every irreducible represen tation has a highest weight.

2. Two irreducible representati ons with the same highest weight are equivalent.

3. The highest weight of every irreducible representation is a dominant integral element.

4. Every dominant integral element occurs as the highest weight of an irreducible represen-

tation.

3.8 The Weyl character formula

Let Σ be a ﬁnite-dimensional irreducib l e rep r esentation of K acting on a vector space V ,

then we co n si d er the space of matrix entries of Σ. Suppose we choose a basis {u

} for V .

Then, for each x ∈ K, the linear operator Σ(x) can be expressed as a mat r i x with respect

to this basis; we d en ot e the entries of this matrix as Σ

kℓ

. Then, a mat ri x entry for Σ is a

function on K that can be expressed in the form

f(x) =

dim V



k,ℓ=1

kℓ

Σ(x)

kℓ

, (3.6)

for some set of c on st a nts a

kℓ

. We can describe the space of matrix entries in a basis-

independent way as the space of fu nctions that can be expressed in the for m

f(x) = tr(Σ(x)A) = t r ace( Σ( x) A) , (3.7)

for some linear operator A on V . To see the equivalence of these two forms, let A

kℓ

be the

matrix for t he operator A in the basis {u

}. Then, the matrix for Σ(x)A is given by the

matrix product (Σ(x)A)

kℓ



Σ(x)

mℓ

, so

tr(Σ(x)A) =

dim V



k,m=1

Σ(x)

mℓ

Thus, every function of the form (3.7) can be expressed in form (3.6) with α

kℓ

= A

ℓk

, and

conversely.

Deﬁnition 3.8.1. Let K be a simply-connected compact Lie group and let Σ be a ﬁnite-

dimensional irreducible representation of K. Then, the character of Σ is the function on

K given by

char Σ(x) = tr(Σ(x)) = tr ac e( Σ(x)).

This function is a matrix entry, ob t ai n ed by taking A = I in (3.7) or tak ing α

kℓ

= δ

kℓ

(3.6). The character is special because it satisﬁes

tr(Σ(xyx

−1

)) = tr(Σ(x)Σ(y)Σ(x)

−1

) = tr(Σ(y)),

for all x, y ∈ K. Rec al l from Section 2.1 that any function f satisfying f(xyx

−1

) = f(y),

∀x, y ∈ K, is called a class function (constant on each conjugacy class of K). Onl y as

a note, the Peter-Weyl Theorem states that th e family of class functions tr Σ forms an

orthonormal basis for L

(K, µ), wher e Σ ranges over the equivalence classes of irreducible

ﬁnite-dimensional representations of K.

We now assume that K is a simply -co n n ec te d compact Lie group (there is also a version

of the result for connected com p a ct Lie groups that are not simply connected). We choose,

as usual, a maximal commutative subalgebra t of k an d we let T be the connected Lie

subgroup of K whose Lie algebra is t. It can be shown that T is a closed subgroup of K

(called a ‘maximal t or u s’ ) . It can further be shown that every el em ent of K is conju ga t e

to an element of T . This means that the values of a class function on K are, in principle,

determined by its values on T . The Weyl character formula is a formula for the restriction

to T of the character of an irred ucible representation of K.

We let g denot e the complexiﬁcation of the Lie algebra t of K, so that g is a complex

semisimple Lie algebra. Then , h = t + it is a Cartan subalgebra in g. If we follow dual

notation and regard the roots as elements of h (not in h

∗

), we know that if α ∈ h is a root,

then α, h is imaginary for all h ∈ t, which means that α itself is in it. It is then convenient

to introduce the real roots, which are simply

times the ordinary roots. This means that

a real root is a nonzero element α o f t with the property that there exist s a nonzero x ∈ g

with

[h, x] = iα, hx,

for all h ∈ t (or, equival ently, for all h ∈ h). We can also introduce the real co-roots as the

elements of t of the form h

2α

α,α

, where α is a real root.

In the same way, we will consider the real weights, which we t h i n k of as elements of t in

the same way as for the roots. So, if (Σ, V ) is an irreducible representation of g, then an

element ω ∈ t is call ed a real weight for Σ if there exists a nonzero vector v ∈ V such that

σ(h)v = iω, hv,

for all h ∈ t. Here, σ is the Lie algebra representation associated to the group represen-

tation Σ. An element ω of t is said to be integral if ω, h

 is an integer for each real

co-root h

. (All of the ‘real’ objects are simply

times the corresponding objects without

the qualiﬁer ‘real’). The real weights of any ﬁnite-dimensional representation of g must be

integral. For the rest of this section, all of roots and weight s will be assumed real, even if

this is not explicitly stated.

If α is an integral element, then it can be shown that there is a function f on T satisfyin g

f(e

) = e

iα,h

, (3.8)

for all h ∈ h. Note that because T is connected and commutative, every element t ∈ T can

be exp r essed as t = e

. However, a given t can be expressed as t = e

in many diﬀerent

ways; the content of the above assertion is t hat the right-hand side of (3.8) is independent

of the choice of h for a given t. This means that we want to say that the right-hand side

of (3.8 ) deﬁnes a function on T , not just on t.

Next, we introduce the element δ of t deﬁned to be half the sum of the positive roots:

δ =



α∈Φ

α.

It can be shown that δ is an integral element. (Clearly, ρ = 2δ is integral, but it is not

obvious that δ itself is integral). Finally, if w is any element of the Weyl group W , we think

of w as an orthogonal linear transformation of t in which case, ǫ(w) = det(w) = ±1. We

are now ready to state the Weyl character formula [91, p.213].

Theorem 3.8.2 (Weyl Character Formula). If Σ i s an irreducible representation of K with

highest real weight ω, then we have

char Σ(e

) =



w∈W

ǫ(w)e

iw·(ω+ρ),h



w∈W

ǫ(w)e

iw·ρ,h

for all h ∈ t, for which the denominator of right-hand side above is nonzero. Or equivalently,





w∈W

ǫ(w)e

iw·ρ,h



char Σ(e

) =



w∈W

ǫ(w)e

iw·(ω+ρ),h

for all h ∈ t. Here, ρ denotes the sum of the positive real roots.

The set of points h for which the denominato r of the Weyl character formula, the so-called

Weyl denominator, is nonzero is dense in t. At points where the denominator is zero, there

is an apparent singularity in th e formula for char Σ. However, actually at su ch points the

numerator is also zero and the character itself rema i n s ﬁnite (as must be the case since,

from the deﬁnition of the character, it is well deﬁned and ﬁnite at every point). Note that

the character formula gives a formula for the restriction of char Σ to T . Since char Σ is a

class function and since (as we have asserted but not p r oved) every element of K is conju-

gate t o an element of T , knowing char Σ on T determin es, char Σ on all of K. A sketch of

the pr oof of the Weyl character formula can be found in [22] or [91].

In fact, the expression



ǫ(w)e

iw·ω,h

appearing on denominator of the Weyl character

formula can be written in an alternate way. This is established in the next r esult [116,

p.264]

Theorem 3.8.3 (Weyl Denominator Identity). On the same assumption of Weyl character

formula, we have



w∈W

ǫ(w)e

iw·ρ,h

= e



α∈Φ

(1 − e

3.9 Representations of SU(3)

As an illustration of the con cepts introduced in this chapter, we will discuss the special

case of the representation theory of SU(3). The main result we have done in this chapter

is Theorem 3.7.5, which states th at an irr educible ﬁnite-dimensional representation of a

semisimple Lie al geb r a can be classiﬁed in terms of its highest weight.

The group SU(3) is simply connected , and so the ﬁnite-dim ensional representations of

SU(3) are in one-to-one cor r espondence with the ﬁnite-di m en s io n al representations of the

Lie algebra su(3). Meanwhile, the complex representations of su(3) are in one-to-one

correspondence with the complex-linear representations of th e complexiﬁed Lie algebra

(su(3))

= sl

only if the associated representation of su(3) is irreducib le, and this holds if and on l y if the

associated complex-linear representation of sl

sition 3.1.5, Proposition 3.1.6, and th e connectedness of SU(3)). This correspondence is

determined by the proper ty that

Π(e

) = e

π(x)

, for all x ∈ su(3) ⊆ sl(3).

Since SU(3) is compact, Proposition 3.2.5 tells us that all of the ﬁnite-dimensional repre-

sentations of SU(3) are direct sums of ir r ed u c ible representations. The above paragraph

then implies that the same holds for sl

(C), that is, sl

property. We can apply th e same reaso n i n g to th e simply-connected group SU(2), its Lie

algebra su(2), and it s complexiﬁed Lie algebra sl

(C). Thus, every ﬁnite-dimensional rep-

resentation of sl

We will use the following basis for sl

(C):





1 0 0

0 −1 0

0 0 0





, h





0 0 0

0 1 0

0 0 −1





;





0 1 0

0 0 0





, x





0 0 0

0 0 1

0 0 0





, x





0 0 1

0 0 0





;





0 0 0

1 0 0

0 0 0





, y





0 0 0

0 1 0





, y





0 0 0

1 0 0





Note that the span of {h

, x

, y

} is a subalgebra of sl

(C)

(Example 3.1.10) by ignoring the third row and column in each matrix. Similarly, the

span of {h

, x

, y

} is a subal gebra isom or p hic to sl

(C). Thus, we h ave the following

commutation relations:

, x

] = 2x

, [h

, x

] = 2x

, y

] = −2y

, [h

, y

] = −2y

, (3.9)

, y

] = h

, [x

, y

] = h

We now list all of the commutation relations among th e basis elements which involve at

least on e of h

and h

(this includes some repetition s of the above commutation relat i on s ) .

, h

] = 0;

, x

] = 2x

, [ h

, y

] = −2y

, x

] = −x

, [ h

, y

] = y

;

, x

] = −x

, [ h

, y

] = y

, x

] = 2x

, [ h

, y

] = −2y

;

, x

] = x

, [ h

, y

] = −y

, x

] = x

, [ h

, y

] = −y

Finally, we list all of the remaining commutation relations.

, y

] = h

, [ x

, y

] = h

, y

] = h

+ h

;

, x

] = x

, [y

, y

] = −y

, y

] = 0, [x

, y

] = 0;

, x

] = 0, [y

, y

] = 0,

, x

] = 0, [y

, y

] = 0;

, y

] = y

, [ x

, y

] = x

, y

] = −y

, [ x

, y

] = −x

All of the an a ly si s we will do for the representations of sl

basis. From now on, all representations of sl

and com p l ex linear.

Now, den ot e by h the com p l ex subal gebra generated by the element s {h

, h

}. Observe

from the relations listed above that:

(i) [ h

, h

] = 0;

(ii) [x

, h

] = 0 and [y

, h

] = 0, for i = 1, 2, 3 and j = 1, 2;

(iii) The oper a t or s ad h

and ad h

are di ago n al i zable.

Thus, the algebra h satisﬁes the conditions (i ) , (ii) and (iii) of Deﬁni t io n 3.3.1, so h is a

Cartan subalgebra of sl

(C).

Recall from Section 3.7 that a weight for a representation π of the algebra sl

to the Cartan subalgebra h) is an element ω ∈ sl

satisfying

π(h)v = ω, hv, for all h ∈ h.

If we denote m

= ω, h

 and m

= ω, h

, then we can see ω as the ordered pair

ω = (m

, m

), and the above condition says that ω is a weight if

π(h

)v = m

v, π(h

)v = m

v. (3.10)

A nonzero vector v satisfying relations (3.10) is called a weight vector corresponding t o the

weight ω = (m

, m

). Recall also that the multiplicity of ω a weight is th e di m en si on of

the cor r esponding weight sp ac e, i. e. the space of all vectors v satisfying (3.10). Thus, a

weight is simply a pair of simultaneous eige nvalues for π(h

) and π(h

). It can be shown

that equivalent repr es entations have the same weights and multiplicities.

Here is the advantage of work with the c om p l ex iﬁcation of Lie algebras: since we are

working over the complex numbers, π( h

) has at lea st one eigenval u e m

∈ C. If W ⊆ V

is the eigenspace for π(h

) with eigenval u e m

, since [h

, h

] = 0, π(h

) commutes with

π(h

), and, so, π(h

) must map W into itself. Thus, π(h

) can be viewed as an operator

on W, and its restriction of to W must have at least one eigenvector w with eigenvalue

∈ C, giving w a si multaneous eigenvector for π(h

) and π(h

) with eigenvalues m

and

, respectively. He n ce, we have

Proposition 3.9.1. Every representation π of sl

Now, every representation π of sl

the subalgebra {h

, x

, y

}

∼

(C). Note that even if π is irreducible as a repre sentation

of sl

(C), there is no reason to expect that it will still be irreducible as a representation of

the subalgebra {h

, x

, y

}. Nevertheless, π restricted to {h

, x

, y

} must be some ﬁnite-

dimensional representation of sl

(C). The same reasoning app l i es to the rest r ic ti o n of π to

the su b al gebra {h

, x

, y

}, which is also isomorphic to sl

(C). Now, recall Theorem 3.1.11,

which tells us that in any ﬁnite-dimensional representation of sl

(C), irr educible or not, all

of the eigenvalues of π(h) must be integers. Applying this result to the restriction of π to

, x

, y

} and to the restriction of π to {h

, x

, y

}, we can state the foll owing corollary

Corollary 3.9.2. If π is a representation of sl

(C), then all of the weights of π are of the

form ω = (m

, m

), with m

and m

being integers.

Recall now that a root of sl

functional α on h such that there exists a nonzero element z of sl

[h, z] = α(h)z, for all h ∈ h.

We can see α as an ordered pair α = (a

, a

) ∈ C

, and the above condit i on says that a

nonzero α is a root of sl

, z] = a

z, [h

, z] = a

z. (3.11)

The element z satisfying rel at i on s (3.1 1 ) is called a root vector corresponding to the roo t

α = (a

, a

). Thus, z is a simultaneous eigenvec to r for ad h

and ad h

. This means that z

is a weight vector for the adjoint rep r esentation with weight (a

, a

). Taking into account

the nonzero condition for (a

, a

), we may say that the roots are prec is ely the nonzero

weights of the adjoint representation. Corollary 3.9.2 then tells us that for any root, both

and a

must be integers, which we can also see dir ect l y in Table 3.3. The commutation

relations (3.9) t el l us what the roots for sl

Note that h

and h

are also simultaneous eigenvectors for ad h

and ad h

, but th ey are not

root vectors because the simultaneous eigenvalues are both zero. Si n ce the vectors in Table

3.3 together with h

and h

form a basis for sl

(C), it is not hard to show that the roots

listed in Table 3.3 are the only roots . These six roots form a root syst e m , conventionally

root α root vector z

(2, −1) x

(−1, 2) x

(1, 1) x

(−2, 1) y

(1, −2) y

(−1, −1) y

Table 3.3: Roots for the Lie algebra sl

(C).

called A

. It is convenient to single out a basis ∆ consis t ing on the two roots correspon ding

to x

and x

and give them special names:

= (2, −1),

= (−1, 2). (3.12)

The roots α

and α

are called the posit ive simple roots, because they have the property

that all of the roots can be expressed as linear combinations of α

and α

with integer

coeﬃcients, and t h e se coeﬃcients are (for e ach root) either all nonnegative or nonposi t i ve.

This is veriﬁed by direct computation:

root α linear combination z

(2, −1) α

(−1, 2) α

(1, 1) α

+ α

(−2, 1) −α

(1, −2) −α

(−1, −1) −α

− α

Table 3.4: Roo t s for sl

, α

The decision to designate α

and α

as the positive simple roots is arbitrary; any other pair

of roots with similar properties would do ju st as well. For examp l e, if we set α

= α

+ α

then we can deﬁne ∆

′

= {α

, α

} as another basis. Figure 3.1 shows the root system for

(C).

The signiﬁcance of the roots for the representation theory of sl

following lemma.

root α linear combination z

(2, −1) α

(−1, 2) −α

− α

(1, 1) α

(−2, 1) −α

(1, −2) α

+ α

(−1, −1) −α

Table 3.5: Roo t s for sl

′

= {α

, α

Figure 3. 1: The root system A

for th e Lie algebra sl

(C).

Lemma 3.9.3. Let α = (a

, a

) be a root and z

a corresponding root vector in sl

(C). Let

π be a representation of sl

(C), ω = (m

, m

) a weight for π, and v = 0 a corresponding

weight vector. Then,

π(h

)π(z

)v = (m

+ a

)π(z

)v,

π(h

)π(z

)v = (m

+ a

)π(z

)v.

Thus, either π(z

)v = 0 or π(z

)v is a new weight vector with weight ω + α = (m

, m

+ a

Proof. In fact, the deﬁnition of a root tells us that we have the commutation relation

, z

] = a

. Thus,

π(h

)π(z

)v =



π(z

)π(h

) + a

π(z

)



v = π(z

)(m

v) + a

π(z

= (m

+ a

)π(z

)v,

and a similar argument allows us to comput e π(h

)π(z

)v.

We see then that if we have a representation with a weight ω = (m

, m

), then by applying

the root vectors x

, x

, y

and y

, we can get some new weights of th e form ω + α,

where α is the root. O f course, some of the time π(z

)v will be zero, in which case ω + α is

not necessarily a weight. In fact, since our repres entation is ﬁnite di m ensional, there can

be only ﬁnitely many weights, so we must get zero quite often. Now we would like to single

out in each representation a highest weight. Recall from a pr evi ous section that if ω

and

are two weights, then ω

is higher than ω

(ω

≻ ω

) if ω

− ω

can be written in the

form

− ω

= a

+ a

, wit h a

, a

≥ 0,

and recall that maximal elements of this pa r t ia l relation are called highest weights. Note

that the relation of ‘hi gher’ is only a partial ordering; th at is, one can easil y have ω

and ω

such that ω

≻ ω

neither ω

≺ ω

. For example, α

− α

is neither higher nor lower than

0. This, in particular, means that a ﬁnit e set of weights need not have a hi ghest element

(e. g., t he set {0, α

−α

} h as no highest element). Note also that the coeﬃcients a

and a

do not have to be integers, even if both ω

and ω

have integer entries. For example, (1, 0)

is higher than (0, 0) since (1, 0) − (0, 0) = (1, 0) =

. Recall also that an ordered

pair (m

, m

) with m

and m

being non-negative integers i s called a dominant integral

element.

Figure 3.2: Root s and dominant integral elements for sl

(C).

Theorem 3.7.5 tell us t hat the highest weight of each irre d u ci b l e representation of sl

(C)

is a dominant integral element and, conversely, that every dominant integral el em ent oc-

curs as the highest weight of some irreducible representation. Since (1, 0) =

and (0, 1) =

, we see that every dominant integral element is high er than zero.

However, if ω has integer coeﬃcients and is higher than zero, this does not necessarily

mean t h at ω is dominant integral (for example, α

= (2, −1) is higher than zero, but is

not dominant integral). Figure 3.2 shows the roots and dominant integral elements for

(C). This picture is made using the obvious basis fo r the space of weights; that is, the

x-coordinate is the eigenval u e of h

and the y-coordinate is the e ig envalue of h

It is possible to obtain much more informat i on about the irreducible repr esentations besides

the high e st weight. For example, we have the foll owing formula for the di m en si on of the

representation with highest weight (m

, m

). It is a consequence of the Weyl character

formula (see [100]):

Theorem 3.9.4. The irreducible representation with highest weight (m

, m

) has dimension

+ l)(m

+ 1)(m

+ m

+ 2).

There is an import ant symmetry to the representations of sl

To understand the idea behi n d the Weyl group symmet r y, let us observe th at the repr e-

sentations of sl

This mea n s the following: let π be a ﬁnite-dimensional representation of sl

a vector space V and let Π be the associated representation of SU(3) acting on the same

space. For any A ∈ SU(3), we can deﬁne a new representation π

of sl

(C), acti n g on the

same vector space V , by setting

(x) = π(AxA

−1

Since the adjoint action of A on sl

representation of sl

(C). Thi s new representation is to be equival ent to the original repre-

sentation; and direct calculation shows that Π(A) is an intertwining map between (π, V )

and (π

, V ) . We may say, th en , tha t the adjoint action of SU(3) is a symmetry of the set

of equivalence classes of representations of sl

(C).

Now, we have analyzed the representations of sl

operators π(h

) and π(h

). Of course, this means that any lin ear combination of π(h

) and

π(h

) is also simultaneously diagonalized. So, what really count s is the two-dimensional

subspace h of sl

and h

, the Car t an subalgebra of sl

(C). In general,

the adjoi nt action of A ∈ SU(3) does not preserve the space h and so the equival ence of

π and π

does not (in general) tell us anything about the weights of π. However, there

are elements A ∈ SU(3) for which ad A does preser ve h. We have already seen that these

elements make up the Weyl group for SU(3) and give rise to a symmetry of the set of

weights of any representation π. So, we may say that the Weyl group is the ‘ r esi d u e ’ of the

adjoint symmetry of the representations (discussed in the previous para gr ap h) that is l eft

after we focus our attention on the Cartan subalgebra h of sl

(C).

In the case of the two-dimension a l subspace of sl

and h

, let Z be the

subgroup of SU(3) consisting of those A ∈ SU(3) such that Ad A(h) = h, for all h ∈ h.

Let N be the subgroup of SU(3) consisting of those A ∈ SU(3) such that Ad A(h) is an

element of h, for all h ∈ h. In fact , Z and N ar e actually subgroups of SU(3) and Z is

a normal subgroup of N. This leads us to the Weyl group W = N/Z of SU(3). We can

deﬁne an action of W on h as follows. For each element w of W , choose an element A of

the corr esponding equivalence class in N. Then for h ∈ h we deﬁne the action w · h by

w · h = Ad A(h).

To see that this action is well deﬁned, suppose B is another element of the s am e equivalence

class as A. Then B = AC, with C ∈ Z and thus, Ad B(h) = Ad AC(h) = Ad A Ad C(h) =

Ad A(h), by the deﬁnition of Z. It can be proved that W is isomorphic to the group of

linear transformations of h that can be expressed as Ad

for so m e A ∈ N. In fact, the

group Z consists pr eci sel y of the diagonal matrices insid e SU(3), namely the ma t r ic es of

the form

A =





iθ

0 0

0 e

iφ

0 0 e

−i(θ+ φ)





, for θ, φ ∈ R.

The group N consists of precisely those mat r i ces A ∈ SU(3) such that for each k = 1, 2, 3

there exist ℓ ∈ {1, 2, 3} and φ ∈ R such that Ae

= e

iθ

ℓ

. Here, {e

, e

} is the standard

basis for C

. Hence, th e Weyl group W = N/Z is iso m or p hic to the permutation group on

three el em ents (see [91] for a proof).

In the case of SU(3), it is pos si b l e to identify the Weyl group with a cert ai n subgroup

of N , instead of as the quotient group N/Z. We want to show that the Weyl group is a

symmetry of the weights of any ﬁnite-dimensional representation of sl

(C). To understand

this, we adopt a less basis-dependent view of the weights. We have deﬁ n e d a weight as a

pair (m

, m

) of simultaneous eigenvalues for π(h

) an d π(h

). However, if a vector v is an

eigenvector for π(h

) and π(h

) then it is also an eigenvector for π(h) for any element h of

the space h span n ed by h

and h

. Furthermore, the ei genvalues must depend lin ear l y on

h since if h and j are any two elements of h and π(h)v = λ

v and π(j) = λ

v, then

π(ah + bj)v = (aπ(h) + bπ(j))v = (aλ

+ bλ

)v.

So, we may make the following basis-independent notion of a weight.

Deﬁnition 3.9.5. Let h be the subspace of sl

and h

and let π be a

ﬁnite-dimensional representation of sl

µ ∈ h is called a weight for π if there exists a nonzero vector v ∈ V such that π(h)v = µ(h)v,

for all h ∈ h. S u ch a vector v is called a weight vector with weight µ.

So, a weight is just a col le ct io n of simultaneous eigenvalues of all the elements h of h, which

must depend linearly on h and, therefore, deﬁ n e a linear functional on h. Since h

and h

span h, the linear functional µ i s determined by the value of µ(h

) and µ(h

), and thus our

new notion of weight is equivalent to our o ld notion of a weight as just a pair of simultaneous

eigenvalues of π(h

) and π(h

). The reason for adopting this basis-in d ependent approach

is that the action of the Weyl group does not preserve the basis {h

, h

} for h. The Weyl

group is (or m ay be thought of as) a group of linear transformations of h. This m ea n s that

W acts linearly on h, and we denote this action as w ·h. We can deﬁne an ass ociated acti on

on the dual space h

∗

in the following way: For µ ∈ h

∗

and w ∈ W , we deﬁne w · µ to be

the elem ent of h

∗

given by

(w · µ)(h) = µ(w

−1

· h).

The main point of the Weyl grou p from the poi nt of view of representation theory, name ly

that the weights of any representation are invariant under th e a ct i on of the Weyl group.

More expl i cit l y, suppose that π is any ﬁnite-dimensional representation of sl

µ ∈ h

∗

is a weight for π. The n , for any w ∈ W , w · µ is also a weight of h

∗

, and the

multiplicity of w ·µ is the same as the multiplicity of µ. In other words, si n ce the roots are

nothing but the nonzero weights of the adjoint representation, this re su l t tells us that the

roots are invariant under the action of the Weyl group. In order to visualize the action of

the Weyl group, it is convenient to identify h

∗

with h by means of an inner product on h

that is invariant under the action of the Weyl group. Recall that h is a su b space of the space

of diagonal matrices, and we can u se the Hilbert- S chmidt inner prod uct A, B = tr(A

∗

B).

Since the Weyl group acts by permuting the diagonal entries, this inner product (restricted

to th e subspace h) is preserved by the action of W .

We now use this inner prod u c t on h

∗

to identify h. Given any el em ent α of h, the map

h → α, h is a linear fun ct i onal on h (i. e., an element of h

∗

). Every linear fun ct i on a l o n

h can be represented in this way for a unique α in h. Identifying each linear functional

with the corresponding element of h, we will now regard a weight for (π, V ) as a no n zer o

element of h with the property that th er e exists a nonzero v ∈ V such t hat

π(h)v = α, hv,

for all h ∈ h. This is the sa m e as Deﬁnition 3.9.5 except that now, α lives in h an d we write

α, h instead of α(h) on the right. The roots, being weights for th e adjoint representation,

are viewed in a simila r way. Now that the roots and weights live in h instead of h

∗

, we can

use the above inner prod u ct o n h, and with this new point of v i ew the roots α

and α

are

identiﬁed with th e following elements of h:





−1





, α





−1





To check this, we note th at these matrices are indeed in h since the diagonal entries sum

to zero. Then, direct calculation shows tha t α

, h

 = 2, α

, h

 = −1, α

, h

 = −1

and  α

, h

 = 2, in agreement with our earlie r deﬁnit i on of α

and α

in (3 .1 2 ) . So, then,

we can compute the lengths and angles as | |α

= α

, α

 = 2, ||α

= α

, α

 = 2,

and α

, α

 = −1. This means that (with respect to this inner product) α

and α

both

have length

√

2 and the angle θ between them satisﬁes cos θ = −

, so that θ = 120

◦

. We

now consid er the dominant integral elements, which are th e possible highest weights of

irreducible representations of sl

(C). With our new point of view, these are the elements

µ of h such that µ, h

 an d µ, h

 ar e non-n egat i ve integers. We begin by considering th e

fundamental weights µ

and µ

deﬁned by

µ

, h

 = 1, µ

, h

 = 0,

µ

, h

 = 0, µ

, h

 = 1.

These can be expressed in terms of α

and α

as follows:

obtaining





−1





, µ





−2





A calculation then shows that µ

and µ

each have length

√

and that the angle between

them is 60

◦

. The set of dominant integral elements is then precisely the set of linear

combinations of µ

and µ

with non-negat i ve integer coeﬃcients. Note that µ

+ µ

+ α

, an observation that helps i n drawing Figure 3.3 below. Figure 3.3 shows the same

information as Figure 3.2, namely, the roots and t h e dominant integral elements, but now

drawn relative to a Weyl-invariant inner product. We draw only the two-dimen si onal real

subspace of h consisting of those elements µ such that µ, h

 and µ, h

 are real, since all

the roots and weights have this property. In this ﬁ gure, the arrows indicate the roots, the

black dots indicate dominant integral elements (i. e., points µ such that µ, h

 and µ, h



are non-negative integers), an d the triangular grid indicates integral elements (i. e., points

µ such that µ, h

 and µ, h

 are integers).

Let us see how the Weyl group acts on Figure 3.3 . Let (1 2 3) denote the cyclic permutation

of 1,2 and 3, an d let w

(1 2 3)

denote the corresponding Weyl group element. Then, w

(1 2 3)

acts by cyclically permuting th e diagonal entries of each element of h. Thus, w

(1 2 3)

takes

to α

and takes α

to −(α

+ α

). This action is a 120

◦

rotation, counterclockwise in

Figure 3.3. Next, let (1 2) be the permutation that interchanges 1 and 2 and let w

(1 2)

be the corresponding Weyl grou p element. Then, w

(1 2)

acts by interchanging the ﬁrst two

diagonal entries of each element of h and thus takes α

to −α

and takes α

to α

+ α

This correspon ds to a reﬂection about the line perpendicula r to α

. The read er is invited

to calculate the action of the remaining Weyl group elements, and observe that the Weyl

group consists of six elements: the symmetry group of an equilateral triangle centered at

the ori gi n , as indicated in Figure 3.4:

Figure 3.3: Root s and dominant integral elements for sl

Figure 3. 4: The Weyl group for sl

(C).

Chapter 4

Vertex algebras and Kac-Moody

algebras

The notion of a vertex algebras was introduced by Borcherds in [ 8 ] . This is a r i go r ou s

mathematical deﬁnition of the chiral part of a 2-d i m en s io n al quantum ﬁeld t h eor y studied

by physicist since the landmark paper of Belavin, Polyakov and Zamolodchikov [4]. Basi-

cally, vertex algebras are the rigorous formalization of the bosonic theory in mathematical

physics. The main objective of this chapter is to give a quickly understand i n g of what

vertex algebras and vertex operat or algebras are. For a more detailed study, the reader

may r efer to [113], [67] and [7]. In Chapter 9 we shall discuss some other interesting topics

—for our purposes— relat i n g conformal ﬁeld theory and the Moonshine phenomenon.

Subsequently, we will describe h ow vertex algebras arise on the mat h em at i ca l scene, partic-

ularly in the developing of the re p r esentation th eor y of aﬃne Kac-Moody algeb r as. In this

part it will be useful to have in mind some of the results obt a i n ed in the previou s chapter.

The read er may refer to [112], [ 182 ] or [75] for a complete pr esentation of these topics.

4.1 Vertex operator algeb ra s

Deﬁnition 4.1.1. L et F be a ﬁeld. A vertex algebra over F is a vector space V over F with

a collect io n of bilinear maps V × V → V

(u, v) → u

for all n ∈ Z, an d satisfying the following axioms

(i) u

v = 0, for n suﬃciently large, i. e., there exists n

∈ N (depending on u and v) such

that u

v = 0, for all n ≥ n

;

(ii) There exists an element 1 ∈ V such that

−1

v = v, 1

v = 0 for all n = −1,

−1

1 = v, v

1 = 0 for all n ≥ 0;

(iii) (Borcherds’ identity) for all u, v, w ∈ V and m, n, k ∈ Z



i≥0





n+i

m+k−i

w =



i≥0

(−1)







m+n−i

k+i

w) − (−1)

k+n−i

m+i



This disti ngu is he d vector 1 usually appears as |0 in physicist notation. For ex ample see [113].

We will work only with the case F = R. Relation (iii) is called Borcherds’ identity. Since

it is som ewh at reminiscent of the Jacobi identity for Lie algebras (see Deﬁnition 2.3.1), it

is called sometimes the Jacobi identity for vertex algebras.

Denote by End V [[z, z

−1

]] th e set of formal series ( i n the indeterminat e z) of the for m



n∈Z

−n−1

where ϕ

∈ End V , for all n ∈ Z. For ea ch u ∈ V , we can deﬁne the vertex operator

Y (u, z) : V → V by

Y (u, z) =



n∈Z

−n−1

, (4.1)

where u

∈ End V is given by v → u

v. We denote by V [[z]] the set of all formal series



i≥0

, for v

∈ V . Also, consider the formal expression

δ(z − w) = z

−1



n∈Z





∈ F[[z, z

−1

, w, w

−1

]].

Then, the axioms for a vertex algebra can be written in terms of such vertex operators as

follows:

(i) Y (u, z)v has coeﬃcient of z

equal to 0 for all n suﬃciently small (i. e., there exists

∈ N dependi n g on u and v such that u

v = 0, ∀n ≥ n

);

(ii) There exists an element 1 ∈ V such that

• Y (1, z) ≡ 1

is the identity on End V ,

• Y (v, z)1 ∈ V [[z]], for all v ∈ V ,

• lim

z→0

Y (v, z)1 = v.

(iii) δ(z

−z

)Y (u, z

)Y (v, z

) −δ(z

−z

)Y (v, z

)Y (u, z

) = δ(z

−z



Y (u, z

)v, z



, for

all u, v ∈ V .

For a series a(z) =



∈ V [[z, z

−1

]], we will denote by

∂a(z) =



n−1

the form al derivat i ve of a(z). Let T : V → V the linear map deﬁned by

T (v) = v

−2

Denote by [·, ·] the usu a l bracket [a, b] = ab − ba deﬁned in End V [[z, z

−1

]]. No t e that if

[T,



] =



[T, a

, for an element



∈ End V [[z, z

−1

]], t h e n

[T, Y (u, z)] =





−n−1





[T, u

−n−1



(T u

− u

T )z

−n−1

. (4.2)

Also,

∂Y (u, z) = ∂



−n−1



(−n − 1) u

−n−2



(−n)u

n−1

−n−1

. (4.3)

On the othe r hand, i f we take w = 1, and m = 0, k = −2 on Borcherds’ identity, we obtain



i≥0





n+i

−2−i

1 =



i≥0

(−1)







n−i

−2+i

1) − ( −1 )

−2+n−i





i≥0

(−1)





n−i

−2+i

1).

Note that t h e sum on the left-hand sid e above involves only one term (when i = 0), and the

right-hand side actual l y has only two terms (when i = 0, 1), since v

1 = 0, for all n ≥ 0.

In fact, we have −nu

n−1

v = (u

−2

1 − u

−2

1) and it follows that

−nu

n−1

v = (u

−2

1 − u

−2

1) = (T u

− u

T )v, for all n ∈ Z. (4.4)

From equation ( 4. 2 ) , (4.3) and (4.4), then we deduce that −nu

n−1

= [T, u

], and hence

[T, Y (u, z)] = ∂Y (u, z). We summarize th i s and other similar results in the next s t at em ent.

See [113 , p.117–118] for a complete proof of this fact.

Theorem 4.1.2. We have the following equivalent axioms for a vertex algebra:

(i’) (translation covariance) [T, Y (u, v)] = ∂Y (u, z), for all u ∈ V ;

(ii’) (vacuum) Y (1, z) = 1

and Y (u, z)1|

z=0

= u, for all u ∈ V ;

(iii’) (locality) (z − w)

Y (u, z)Y (v, w) = (z − w)

Y (v, w)Y (u, z) for n suﬃciently large

(depending on u and v).

Now, we apply the operator T rep e at ed l y to the equation T v = v

−2

1. Since T v

[T, v

] + v

T , we have that

T (v

1) = (−n)v

n−1

1 + (v

T )1 = (−n)v

n−1

1 + v

−2

1) = (−n)v

n−1

1, (4.5)

and so

v = T (v

−2

1) = 2v

−3

1 so that v

−3

1 =

v = T (2v

−3

1) = (2 · 3)v

−4

1 so that v

−4

1 =

Continuing this process on identity (4. 5 ) , an induction on n guar antees t hat v

−n

1 =

(n−1)!

n−1

v, for all n ≥ 1. Hence

Y (u, z)1 =



1)z

−n−1



n≥0

1)z

−n−1



n<0

1)z

−n−1



n≥0

−(n+1)

1)z



n≥0

u z





for all u ∈ V .

Remark 4.1.3. The bracket operation deﬁned by [u, v] = u

v makes V/T V into a Lie

algebra (see [12] for a proof of this). Borcherds’ identity is described in [67] as being ‘very

concentrated’. It can be shown that it is equivalent to three simpler identities [113 ]. For

θ, ϕ ∈ End V , we deﬁne [θ, ϕ] = θϕ −ϕθ. Also, if a(z) =



−n−1

∈ End V [[z, z

−1

]], we

deﬁne

a(z)



n<0

−n−1

= a

−1

+ a

−2

z + a

−3

+ . . .

a(z)

−



n≥0

−n−1

= a

−1

+ a

−2

+ a

−3

+ . . .

Deﬁnition 4.1.4. Given a(z), b(z) ∈ End V [[z, z

−1

]], we also deﬁne t h ei r normal ordered

product as

: a(z)b(z) : = a(z)

b(z) + b(z)a(z)

−

In te r m s of the notation just introduced, Borcherds’ identity can be shown t o be equivalent

to th e following three simpler identities:

(a) [u

, Y (v, z)] =



i≥0





Y (u

v, z)z

m−i

, for all u, v ∈ V , m ∈ Z;

(b) : Y (u, z)Y (v, z) : = Y (u

−1

v, z), for all u, v ∈ V ;

Deﬁnition 4.1.5. An element ω of a vertex algebra V is a conformal vector of central

charge c, if is an even vector satisfying:

• ω

v = T v, for all v ∈ V ;

• ω

ω = 2ω;

• ω

ω = 0;

• ω

ω =

• ω

ω = 0, for all i ≥ 4;

• V =



n∈Z

, where each V

is the set of eigenvectors of the line ar operat or ω

= {v ∈ V : ω

v = nv}

corresponding to the eigenvalue n ∈ Z.

In oth er words, ω is a conformal vector if the corresponding vertex operator Y (ω, z) is a

Virasoro ﬁeld with central charge c, i. e., a formal series L(z) satisfying

L(z)L(w) =

c/2

(z − w)

2L(w)

(z − w)

∂L(w)

z − w

In particular, for a conformal vector ω we have, 1 ∈ V

and ω ∈ V

. Note th a t wh en

a vertex algebra V has a conformal vector, it admits an action of the Lie algebr a called

Virasoro algebra ( see next section).

Deﬁnition 4.1.6. A vertex algebra en d owed with a conform al vector ω as in Deﬁnition

4.1.5 i s called a vertex operator algebra (or a conformal vertex algebra) of rank c.

4.2 The Virasoro algebra

Let p(t) ∈ F[t, t

−1

] and consider the derivation

p(t)

= p(t)∂ (4.6)

of F[t, t

−1

]. The linear space of all derivations of F[ t, t

−1

] of type (4.6) has th e structure of

a Lie algebra with respect to the natural Lie bracket

p(t)

, T

q(t)

] = T

p(t)q

′

(t)−p

′

(t)q(t)

for p(t), q(t) ∈ F[t, t

−1

]. We denote this algebra by d and we choose th e following basis of

= −t

n+1

∂ = −t

d : n ∈ Z},

where d is t h e derivation in (2.12). Then, the commutators have the form [d

, d

] =

(m − n)d

m+n

, for m, n ∈ Z.

In fact, if T ∈ End F[t, t

−1

] is a derivation, and we set

p(t) = T (t), (4.7)

then we have T (1) = T (1 · 1) = T ( 1) + T (1), so that

T (1) = 0, (4.8)

and 0 = T (tt

−1

) = T (t)t

−1

+ tT (t

−1

) impl i es that

T (t

−1

) = −t

−2

T (t). (4.9)

Since formulas (4.7), (4.8) and (4.9) also hold for T

p(t)

, we see that T

p(t)

and T agree on all

powers of t. Thus, we have the following statement explaining the importance of d.

Proposition 4.2.1. The derivations of F[t, t

−1

] form precisely the Lie algebra d.

Any th r ee generators of d of t h e form d

, d

−n

, n ∈ Z

span a subalgeb r a of d isomorphic

to the Lie algebra sl

(F) of 2 × 2 matrices over F of trace 0. We shall single out the

subalgebra

p = Fd

+ Fd

−1

As in the case of aﬃne Lie algebras, we consider central extensi ons. We denote by v the

following one-dimensional central extension of d with basis con si st i n g of a central element

c and elements L

, n ∈ Z, corresponding to the basi s elements d

, of d. For m, n ∈ Z

, L

] = (m − n)L

m+n

− m)δ

m+n,0

c (4.10)

= (m − n)L

m+n



m + 1



m+n,0

c; (4.11)

and [L

, c] = 0, fo r all m ∈ Z.

Deﬁnition 4.2.2. The Lie algebra d above is called the Virasoro algebra.

Note th at the central extension (4.10) is trivial wh en restricted to the subalgebra p of d.

We can form equivalent extensions of d by setting

′

= L

+ β

c, fo r β

∈ F, n ∈ Z. (4.12)

Then, the exten si on (4.10) is modiﬁed by the subtraction of the term (m − n)β

m+n

Although, the signiﬁcance of the extension (4.10) is given by the next result (see [67, p.33]

for a proof).

Proposition 4.2.3. The extension (4.10) of the Lie algebra d is the unique nontrivial

1-dimensional extension up to isomorphism.

Now we return to the situation in which V is a vertex operator algebra with conformal

vector ω. Deﬁne the l i n ea r map L

: V → V by

= ω

n+1

, for n ∈ Z.

The pr operties of ω imply t h a t

, L

] = (m − n)L

m+n



m + 1



m+n,0

c 1

, (4.13)

in analog y to (4.11) (see [113] for details). Thus, V is a module for the Virasoro algebra in

which th e central element c i s represented by c 1

, where c is the central charge of ω.

4.3 Kac-Moody algebras and their representa t io n

We shall now describe how vertex operators arose in the representation theory of aﬃne

Kac-Moody algeb r as . In order to explain th is we ﬁrst recall some ideas from the th eo r y of

Lie alg eb r as (see Chapter 3).

We su p pose ﬁrst that g is a ﬁni t e dimen s io n al sim p l e Lie algebra over C. Then, g has a

decomposition

g = n

⊕ h ⊕ n

−

, (4.14)

where



α∈Φ

, n

−



α∈Φ

−

and all g

satisfy

dim g

= 1 and [h, g

] = g

Here, h is a Cartan subalgebra of g, and the 1-dimensional spaces g

are the root spaces of

g with respect to h, each of which gives rise to a 1-dimension a l representation α of h gi ven

[x, x

] = α(x) · x

, for all x

∈ g

, x ∈ h.

Recall that the set Φ = Φ

∪Φ

−

is the set of roots of g. Φ contains a subset ∆ = {α

, . . . , α

}

of simple roots, where r = di m h. The roots in Φ

are linear combinations of elements o f ∆

with non-negative integer coeﬃcients and roots in Φ

−

are linear combinations of eleme nts

of ∆ with non-positive integer coeﬃcients. The free abeli an group

Q = Zα

+ Zα

+ . . . + Zα

is called the root lattice. The real vector space Q ⊗

R can be given the str u ct u r e of a

Euclidean space in a natural way. Let w

be t h e reﬂection in the wall orthogonal to α

Then, we h ave seen that the group of isometri es of Q ⊗ R generated by w

, . . . , w

is the

Weyl gr oup W of g. Reme mber that this is a ﬁnite group which permutes the elements of

Φ (Proposition 3.5.2). Each root is the image of some si m p l e root under an element of W .

We have

(α

) = α

− A

, for A

∈ Z. (4.15)

Deﬁnition 4.3.1. The matrix A = [A

] is called the Cartan matrix of g (relatively to the

Cartan subalgebra h).

Any Cartan matrix [A

] satisﬁes the following conditions:

• A

= 2, for each i;

• A

∈ {0, −1, −2, −3} if i = j;

• A

= 0 if and only if A

= 0;

• A

∈ {−2, −3} ⇒ A

= −1.

The Lie algebra g can be deﬁn e d by generators and rel at i ons depen d i n g only on the Cartan

matrix A. The Cartan subalgebra h has a basis h

, . . . , h

of elements satisfying

) = A

We also recall h ow to describe the ﬁ n i t e dimensional irreducible g-modules. These are

parametrized by dominant integral weights. Remember th at a weight is an element of

∗

. A weight ω is said dominant if ω(h

) is a non-negative real number for all i. Finally,

ω is dom i n ant integral if each ω(h

) is a non - n eg at i ve integer. The fund a m ental weights

, . . . , ω

are deﬁ n ed by

) = δ

, for i, j)1, 2, . . . , r.

Thus, each dominant integral weight λ is a non-negati ve integral combination of the fun -

damental weights.

For each dominant integral weight λ there i s a corresponding ﬁ n i t e dimensional irreducible

g-module M

. Th i s module M

is a d i r ect sum of 1-dimen s io n al h-m odules which, when

collected together, give rise to a weight space decompos it i o n



where M

= {v ∈ M

: xv = µ(x)v, ∀x ∈ h} (here the sum is over all µ ∈ h

∗

). In a simi la r

form to Deﬁnitio n 3.8.1, we deﬁne a not i on of character for this m odule M

Deﬁnition 4.3.2. The character of the h-module M

is the function given by

char M



(dim M

Note that the character deﬁned above is an element of the group ri n g F[G] of the fr ee

abelian group G gen e ra t ed by the fundamental weights. As usual, we write this gr ou p

multiplicatively, replacing a weight µ by e

and su ch that

= e

µ+ν

The Weyl character formula (Theorem 3.8.2) applied to this new deﬁnition of character

asserts t h a t





w∈W

ǫ(w)w(e

)



char M



w∈W

ǫ(w)w(e

λ+ρ

), (4.16)

where ǫ = det ∈ {±1}, is the homomorphism given by ǫ(w

) = −1, for all i, and ρ =



i=1

. In an analo gous way to Theorem 3.8.3, the expression



ǫ(w)w(e

) appearing

in the denominator or char M

can be written in an alternative form. In this case, Weyl’s

denominator identity a sser t s that



w∈W

ǫ(w)w(e

) = e



α∈Φ

(1 − e

−α

). (4.17)

The well known theory of ﬁnite dimensional simple Lie algebras over C which we have just

outlined was generalized by Kac [110] a n d Moody [149] to give the t h eor y of Kac-Moody

algebras. In order to obtain such Lie algebras, we begin with the notion of generalized

Cartan matrix.

Deﬁnition 4.3.3. A generalized Car tan matrix (sometimes abbreviated GCM) is any ma-

trix A = [A

] satisfy ing the conditions

• A

∈ Z;

• A

= 2, for all i;

• A

≤ 0, if i = j;

• A

= 0 if and only if A

= 0.

A Lie algebra is then deﬁned by generators and relations depending on the generalized

Cartan matrix A, just as in the case of ﬁnite dimensional simple Lie algebras. This Lie

algebra is then extended by ou t er derivations to ensure that the simple roots are lin ea rl y

independent, even th ough the matrix A is singular (may occu r ) .

Deﬁnition 4.3.4. The resulting Lie algebra above is called the Kac-Moody algebra given

by the generalized Cartan matrix A.

The mai n diﬀerences from the ﬁnite dimensional case are as follows:

1. The Lie algeb r a g can have inﬁnite dimension.

2. The root spaces g

can have dimension greater than 1.

3. The Weyl group W can be inﬁnite.

4. There can be both real roots and imaginary roots.

In thi s case, a root α is called real if α, α > 0 and imaginary if α, α ≤ 0. All simple

roots of a Kac- Moody algebra are real, and any real root can be obtained from a simple

root just tran sfor m i n g by some appropriate element of the Weyl group.

A general i zed Cartan matrix A is said symmetrisable if A = DB, where B is symmetric and

D is a non-singular diagonal matrix. We shall restrict attention to Kac-Moody algebras

with symmetrisable GCM. For each dominant integral weight λ, t h er e is a cor r esponding

irreducible module M

for such a Kac-Moody algebr a. The character of this module M

given by Kac’s character formula





w∈W

ǫ(w)w(e

)



char M



w∈W

ǫ(w)w(e

λ+ρ

), (4.18)

Of course th i s look very much like Weyl’s character formula (4.16) in the ﬁ n i t e dimensional

case, but here the sums over W are inﬁ n i t e. Th e denominator of Kac’s character formula

can be written in an alternati ve way, giving Kac’s denominator identity (analogous to

(4.17))



w∈W

ǫ(w)w(e

) = e



α∈Φ

(1 − e

−α

)

mult α

. (4.19)

The left-hand side is an inﬁnite sum an d the right-hand side is an inﬁn i t e p roduct. The

multiplicity mult α is given by

mult α = dim g

In the ﬁnite dimensional cas e, all the roots have multiplicity 1 and so equation (4.19 ) re-

duces to Weyl’s denominator identity (4.17).

There is a remarka b l e trichotomy in the theory of Kac-Mood y algebras. Let A be an n ×n

generalized Cartan matrix. Given a vector u ∈ R

we write u ≻ 0 if all coordinates u

of u are positive; and u ≺ 0 if all u

are negative. It turn s out that if the GCM A is

indecomposable, exactly one of the following conditions holds:

• There exist u ≻ 0 with Au ≻ 0;

• There exist u ≻ 0 with Au = 0;

• There exist u ≻ 0 with Au ≺ 0.

We say that A has ﬁnite type if the ﬁrst condition holds; aﬃne type if the second condition

holds, and indeﬁnite type if we have t h e third condi t i on . The indecomposable Kac-Moody

algebras of ﬁnite type corr espond to the ﬁnite dime n si on a l simple Lie algebras. All Kac-

Moody algebras of ﬁni te or aﬃne type are s ym m et r i sable.

Examples of aﬃne Kac-Moody algeb r as are the non-twisted aﬃne a l geb r a s (Section 2.8).

These may be constructed as follows. We start with a ﬁ nite d i m ensional simple Lie algebra

g over C. We then consider the algeb r a g ⊗ C[t, t

−1

]. Thi s algebra has a C-basis

⊗ t

, x

⊗ t

, for i = 1, . . . , r, m ∈ Z,

where x

is a non-zero vector or g

. Also, this algebra h a s a non-trivial 1-dimension al

central ext ension. This is the Lie algebra

g ⊗

C[t, t

−1

] ⊕ Ck,

in which multiplication is given by

[x ⊗ t

+ λk, y ⊗ t

+ λ

′

k] = [ x, y] ⊗ t

m+n

+ x, ymδ

m+n,0

for x, y ∈ g. Here  ·, · is a non-degenerat e invar i ant symmet r i c bi l i n ear for m o n g (r ecal l

equation (2.14)). We have already called this Lie algebra

g the untwisted aﬃne algebr a

associated wi t h g. If we extend this Lie algebra by adjoining a 1-dimensional space of outer

derivations, thus we obtain the Lie algebra

g =

g ⋊ Cd = g ⊗

C[t, t

−1

] ⊕ Ck ⊕ Cd,

with multiplication

[x⊗t

+λk +µd, y ⊗t

+λ

′

k +µ

′

d] = [x, y]⊗t

m+n

−x⊗µ

′

+y ⊗µnt

+x, ymδ

m+n,0

We have called this the extended aﬃn e algebra associated with g (Section 2.8). This Lie

algebra

g is an aﬃne Kac-Moody algebra whose generalized Cart an matrix has degree

n + 1. Its GCM is obtained from the Cartan matrix of g by adjoini n g an additional row

and column. Let α the highest root of the Lie algebr a g and consider α

= −α. The Cartan

matrix of g is the n × n matrix A = [A

], wher e

= 2

α

, α



α

, α



, for i, j = 1, 2, . . . , r;

and the generalized Cartan matrix of t h e aﬃ n e L i e a lg eb r a

g is the (n + 1) ×(n + 1) matrix

A = [A

], for i, j = 0, 1, 2, . . . , r,

given by the same formula. Thus, the GCM of

g gives rise to a corresponding irreducible

g-module M

, as described above. In part i cu l ar , we obtain such modul es associated with

the fun d a m ental weights ω

, ω

, . . . , ω

g. The fundamental representation of

g associ -

ated to the weight ω

is called the basic representation of

g. The basic representation plays

a crucial role in the many application s of the repr esentation theory of aﬃne Kac-Moody

algebras. We sh a ll describe a modul e giving the basic repr ese ntation of

g in the case when

g is simply laced, i. e., when A

= 0 or −1 for all i, j ∈ {1, 2, . . . , r} with i = j.

Suppose this is so and let Q be the root lattic e of g. There i s a symmetric Z-bilinear form

Q × Q → Z given by (α, β) → α, β uniquely determined by the con ditions

α

, α

 = A

, for α

, α

∈ ∆.

This lat t i ce Q has a unique ce ntral extension

Q with

1 −→ C

−→

Q −→ Q −→ 1

in which

aba

−1

= (−1)

¯a,

b

, for a, b ∈

where a → ¯a is the map

Q → Q. For each γ ∈ Q, we choose an element e

∈

Q such that

= γ and e

= 1. Then we have

= e

α+β

ǫ(α, β), for all α, β ∈ Q.

The map ǫ : Q×Q → {±1} is a 2-cocycle. It is known fr om the theory of ﬁnite dimensional

Lie alg eb r as that the root vectors x

of g can be chosen so tha t

, x

] = ǫ(α, β)x

α+β

whenever α, β, α + β ∈ Φ.

We are now in position to describe a

g-module giving the basic r ep r es entation. Let

−

the

subalgebra of

g given by

−



n<0

(h ⊗ t

and let S(

−

) be the symmetric algebra on

−

(recall Section 2.7). Let C[Q] the group

algebra of Q with basis e

for γ ∈ Q. We write

= S(

−

) ⊗

C[Q]. (4.20)

It t u r n s out that V

can be made into a

g-module aﬀording the basic representation. The

central element K of

g acts as the identity ma p on V

. Th e elements h ⊗ t

for h ∈ h act

as follows:

h ⊗ t

acts as



h(n) ⊗ 1 if n = 0,

1 ⊗ h(0 ) if n = 0.

Here, h(0) ∈ End C[Q] is given by h(0)e

= γ(h)e

, and h(n) ∈ End S(

−

) is given by the

following rule

• If n < 0, then h(n) i s multiplication by h ⊗ t

• If n > 0, then h(n) i s the derivation of S(

−

) deter m i n ed by

h(n)(x ⊗ t

−n

) = n(x, h), for x ∈ h,

h(x)(x ⊗ t

−m

) = 0, if m = n.

We now consider the action of the elements x

⊗t

∈

g on V

. Let x

(n) ∈ End V

be the

endomorphism induced by x

∈ t

. The en domorphisms x

(n) turn out to be complicated

expressions not appealing to the intuition. However, it is possible to make sense of them

by combining then into a vertex operator

Y (α, z) =



n∈Z

(n)z

−n−1

which is given by an explicit for mula. In order to explain th i s, we introduce the following

notation.

There is an isomorphism h

∗

→ h given by λ → h

, where µ(h

) = λ, µ. In this way

∗

may be identiﬁed with elements of h ( t he corresponding co-roots). Thus, the elements

α(n) ∈ End S(

−

) are deﬁned as above for n = 0. The vertex operator Y (α, z) is then

given by

Y (α, z) = exp





n<0

−

α(n)

−n



exp





n>0

−

α(n)

−n



Here, z

∈ End V

is given by 1 ⊗ z

for z

∈ End C[Q]; and z

= z

α,γ

. Also,

∈ End V

is given by 1 ⊗e

for e

∈ End C[Q]; and e

= ǫ(α, γ)e

α+γ

(see for example

[112]).

We conclude this section wi t h the deﬁnit i on of automorp h i sm of a vertex algebra. We will

see in Chapter 5 that the Mons t er group M occur as a group of au t o m or p hisms of a vertex

algebra.

Deﬁnition 4.3.5. Let V be a vertex algebra with conformal vector ω. An automorphism

of V is an invertible linear map g : V → V satisfying

gY (v, z)g

−1

= Y (gv, z), for all v ∈ V ;

g(ω) = ω.

This then, is how vertex oper a t or s ﬁrst appeared in the representation theory of aﬃne Kac-

Moody algebras. In fact, as we shall see, the vector space V

can be made into a vertex

algebra.

4.4 Lattices

Deﬁnition 4.4.1. By a lattice of rank n ∈ N we shall mean a rank n free abelian group L

provided with a rational-val u e d symmetric Z-bilin ea r form

·, · : L × L → Q

A lattice isomorphism is sometimes called an isometry. A lattice L is said non-degenerate

if its form ·, · is non -degenerate in the sense that for α ∈ L

α, L implies α = 0.

Given a lattice L, we see by choosing a Z-base of L that L, L ⊆

Z for some r ∈ Z

That is

·, · : L × L →

We canoni cal l y embed L in the Q-vector space

= L ⊗

Q, (4.21)

which is n-dimensional since a Z-b as is of L is a Q-basis of L

, and we extend ·, · t o a

symmetric Q-bilinear form

·, · : L

× L

→ Q. (4.22)

Note that every element of L

is of the form α/N for some α ∈ L and N ∈ Z

. The lattice

L i s non- d eg en er at e if and only if the form (4.22) is non-degenerate, and this amounts to

the con d i t io n

det[α

, α

]

i,j

= 0, (4.23)

for a Z-base {α

, . . . , α

} of L. A lattice may be equivalently deﬁned as the Z-span of a

basis of a ﬁnite-dimension al rational vector space equipped with a symmetric bilinear form.

Let L be a lattice. For m ∈ Q, we se t

= {α ∈ L : α, α = m}. (4.24)

The latti ce L is s ai d to be even if α, α ∈ 2Z, for all α ∈ L. It is said integral if α, β ∈ Z,

for all α, β ∈ L; and is said positive deﬁnite if α, α > 0, for all α ∈ L−{0}, or equivalently,

for α ∈ L

− {0}. The polarization formula

α, β =



α + β, α + β − α, α − β, β



, (4.25)

shows that an even lattice is integral. The dual of L is the set

◦

= {α ∈ L

: α, L ⊆ Z}. (4.26)

This set is again a lattic e if an d only if L is non-degenerat e, and in this case, L

◦

has as a

base th e dual base {α

∗

, . . . , α

∗

} of a given base {α

, . . . , α

} of L, deﬁned by

α

∗

, α

 = δ

for i, j = 1, . . . , n.

Note that L is integral if and only if L ⊆ L

◦

. The lattice L is said to be self-dual if L = L

◦

This is equivalent to L being integral and unimodular, which means th a t



det[α

, α

]

i,j



= 1.

In fact, if L is integral and non-deg en er a te , the [α

, α

]

i,j

is the matrix of the embedding

map L → L

◦

with respect to the given base and its du a l base, and the unimodula r ity

amounts to the condition that this embedding be an isomorphism of abelian groups.

Generalizing (4.21) and (4. 22), we embed L i n the E-vector space

= L ⊗

for any ﬁeld of characteristic zero, and we extend ·, · t o the symmetric E-bilinear form

·, · : L

× L

→ E.

Then, L is positive deﬁnit e if and only if the real vector space L

is an Euclidean space.

In thi s case,

| < ∞, for m ∈ Q, (4.27)

since L

is the intersection of the discrete set L with a compact set (a sphere) in L

Using the Schwarz inequality, we observe that if the lattice L is integral and positive deﬁnite,

and if α, β ∈ L

, then

α, β ∈ {0, ±1, ±2}

and

α, β = −2 if and only if α + β = 0,

α, β = −1 if and only if α + β ∈ L

α, β ≥ 0 if and only if α + β /∈ L

∪ {0}.

Let L be an even lattice. Set

L = L/2L,

and view the elementary abelian 2 -g r ou p

L as a vector space over the ﬁeld F

. Denot e by

L →

L the canonical map α → ˘α = α + 2L. Since a Z-base of L reduces to an F

-basis of

L, the n

dim

L = rank L.

There is a canonical Z-bilinear form c

: L × L → Z/2Z g iven by

(α, β) → α, β + 2Z

on L, an d c

is alternating because L is even. the form c

induces a (well-deﬁned) altern a t ing

-bilinear map c

L ×

L → Z/2Z given by

(˘α,

β) →  α, β + 2Z,

for α, β ∈ L. There i s a canonical quadratic form q

L with associated bilinear form c

L → Z/2Z, with ˘α →

α, α + 2Z, (4.28)

for α ∈ L. this form is well-deﬁned: if ˘α =

β, then β − α = γ ∈ 2L, and

β, β =

α, α + α, γ +

γ, γ ≡

α, α (mod 2).

It is clear that c

is the associated form. Denote by q

: L → Z/2Z, where

α →

α, α + 2Z,

the pu l l b a ck of q

to L. There exist Z-bilinear forms ǫ

: L × L → Z/2Z su ch that

(α, α) = q

(α) =

α, α + 2Z, forα ∈ L,

and con seq u ently

(α, β) − ǫ

(β, α) = c

(α, β) =  α, β + 2Z,

for α ∈ L. Note that q

(or equivalently, c

) is nonsingular if, and only if, the determinant

(4.23) is odd —in p a r t icular, if L is unimodu l ar — . These considerations are important in

the con st r u ct i on of central extensions.

Let L be a positive deﬁnite lattice. The theta function θ

associated to L is deﬁned to be

the form al series in the variable q = e

2πiz

(see Sect i on 6.7) given by

(q) =



α∈L

α,α/2



m∈Q

m/2

, (4.29)

(recall (4.24) and (4.27)). If L is even and uni m odular, the theta function θ

has important

modular transformation properties under the modular group SL

(Z), when z is a comp l ex

variable in th e upper half plane H (see Ch apter 6).

4.5 The vertex algebra of an even lattice

Let L be an even lattice, i. e., a free abelian group of ﬁnite ra n k with a symmetric non-

degenerate bilinear form L × L → Z

(α, β) → α, β

such tha t α, α ∈ 2Z for all α ∈ L.

Example 4.5.1. The root lattice Q of a simply laced simple Lie algebra considered in

Section 4.3 is even.

Let h = L ⊗

C and consider

−



n<0

(h ⊗ t

Let C[L] be the group algebra of L, with basis e

for γ ∈ L. Deﬁne V

by equati on (4.20)

= S(

−

) ⊗

C[L].

then, it is possib l e t o make V

into a vertex algebra with the property that for all α ∈ L

we have

Y (1 ⊗ e

, z) = exp





n<0

−

α(n)

−n



exp





n>0

−

α(n)

−n



where the elements α(n), e

and z

of End V

are deﬁn e d as in Section 4.3 (c. f. [67] or

[113]).

More gen er al l y, for

v = (h

⊗ t

−n

) ···(h

⊗ t

−n

) ⊗ e

∈ V

we have

Y (v, z) = :

− 1)!

(

)

−1

(z) ···

− 1)!

(

)

−1

(z) Y (1 ⊗ e

, z) :

where the normal ordered product of more th an two factors is deﬁn ed inductively from the

right in terms of Deﬁnition 4.1.5, that is, for examp l e

: a

(z)a

(z) : = : a

(z)



: a

(z)a

(z) :



: .

Thus, for h ∈ h and m > 0 we have



(h ⊗ t

−m

) ⊗ 1, z



(m − 1)!





m−1

h(z),

where h(z) =



n∈Z

h(n)z

−n−1

and h(n) ∈ End V

is deﬁned as in Section 4.3. The element

1 ∈ End V

is 1 ⊗ 1. Also, V

has a conformal vector ω given by

ω =





i=1

′

⊗ t

−1

)(h

⊗ t

−1

)



⊗ 1 ∈ V

where h

, . . . , h

is a basis for h, and h

′

, . . . , h

′

is the dual basis with respect to ·, ·. The

element ω is independent of this choice of basis ( c. f. [67]).

The map s L

: V

→ V

deﬁned by L

= ω

i+1

satisfy

, L

] = (i − j)L

i+j



i + 1



dim h δ

i+j,0

Thus, V

is a mod ule for the Virasoro algebra under which the central element c acts as

multiplication by dim h = rank L (see [67]). Since the subspace of the Virasoro algebra

spanned by L

−1

, L

and L

is a 3-dimensional subalgeb r a isomorphic to sl

(C), we may

regard V

as a sl

(C)-module. Then, elements of the 1-dimensional subspace spanned by

1 ∈ V

are ann i h i l at ed by sl

with th i s property.

4.6 Dynkin Diagrams of classic Lie algebras

In this sect i on , we discuss informally (withou t proof) the cl ass iﬁcation, up to equ ivalence,

of root systems. This lead s to a classiﬁcation, up to equivalence, o f semisim p l e Lie algebras.

The classiﬁcation of r oot systems is given in terms of an object called the Dynkin diagram.

Suppose A = {α

, . . . , α

} is a base for a root system R. Then, the Dynkin diagram for

R (relative to the base A) is a graph having vertices v

, . . . , v

. Between any two vertices,

we place either no edge, one edge, two edges, or three edges as follows. Consider distinct

indices i and j. If the correspon d i n g roots α

and α

are orthogonal, th en we put no edge

between v

and v

. In the cases where α

and α

are not orthogona l, we put

• one edge between v

and v

if α

and α

have the same length,

• two edges if the longer of α

and α

√

2 longer than the shorter, and

• three edges if the longer of α

and α

√

3 longer than the shorter.

In addition, if α

and α

are not orthogonal and not of the same length, then we decorate

the edges between v

and v

with an arrow pointing from the vertex associated to the longer

root toward the vertex assoc ia t ed to the shorter root ( thinking of the arrow as a ‘greater

than’ sign). Th ese are all the possible values, since we have [91, p.246–249]

Proposition 4.6.1. Suppose that α, β are roots (of a root system R), α i s n ot multiple of

β, and α, α ≥ β, β. Then, one of the following holds:

1. α, β = 0;

2. α, α = β, β, and the angle between α and β is 60

or 120

;

3. α, α = 2β, β, and the angle between α and β is 45

or 135

;

4. α, α = 3β, β, and the angle between α and β is 30

or 150

Proposition 4.6.2. If α and β are distinct elements of a base ∆ for R, then α, β ≤ 0.

Thus, Propositi on 4.6.1 tells us that if α

and α

are not orthogonal, then the only possible

length ratios are 1,

√

2 and

√

3. Furthermore, Proposition 4.6.2 says that these three cas es

correspond to angles of 120

◦

, 135

◦

, and 150

◦

, respectively.

Two Dynkin diagram s are said to be equivalent if there is a one-to-on e, onto map of the

vertices of one to the vertices of the other that preserves the number of bonds and t he

direction of the arrows. Recall from S e ct io n 3.5 th a t any two bases for the same root

system can be mapped into one another by the action of the Weyl group. This implies

that the equivalence cl ass of the Dynkin diagra m is independent of the choice of base. As

we wil l see, only graph s of certain very special forms arise s as Dynkin di ag r am of a root

system. The following characterizes root systems via their Dynkin diagram [91, p. 27 0]

Theorem 4.6.3. A root system is irreducible if and only if its Dynkin diagram is connected.

Two root systems with equivalent Dynkin diagrams are equivalent. If R

∗

is the dual root

system to R, then the Dynkin diagram of R

∗

is the same as that of R (with direction of

each arrow reversed).

So, t h e classiﬁcation of irreducible root systems amounts to classifying all the connected

diagrams that can arise as Dynkin diagrams of root systems. It is well known the classi-

ﬁcation of the D yn k i n diag r am s for the classical Lie algebras, sl

(C), so

(C), and sp

(C)

(see Figu r e 4.1):

• A

: The root system A

is the root system of sl

n+1

(C), whi ch has rank n.

• B

: The root system B

is the root system of so

2n+1

(C), wh i ch has rank n.

• C

: The root system C

is the root system of sp

(C), wh i ch has rank n.

• D

: The root system D

is the root system of so

(C), whi ch has rank n.

. . .

Figure 4. 1: Dynki n diagram s for A

, B

, C

and D

Certain special things happen in low rank. In r a n k one, t h er e is only one possible Dynkin

diagram, reﬂecting that there is only one isomorphism class of complex semisimple Lie

algebras in rank one. The Lie algebra so

(C), so

(C), and sp

(C), are isomorphic. In r a n k two, the Dynkin diagram D

is discon-

nected, reﬂecting that so

(C)

∼

(C). Also, the Dynkin diagr am s B

and C

are

isomorphic, reﬂecting that so

(C)

∼

(C). In ra n k three, the Dynkin diagrams A

and

are isomorp h i c, thus sl

(C)

∼

(C). We may obser ve certain things about the short

and long roots in root systems where more than one length of root occurs. The lon g roots

in B

form a root system by t hemselves, namely D

. The short roots in B

form a root

system by themselves, na m el y A

× ··· × A

. In C

, it is the reverse: the long roots for m

× ··· × A

and the short roots form D

In ad d i t io n to the root systems associated to the classical Lie algebras, there are ﬁve

‘exceptional’ irreducible root syst em s, denoted G

, F

, E

, and E

, whose Dynkin

diagrams are shown in Figure 4.2. For t h e con st r u ct i on of th es e except i onal root systems,

see Humphreys [100]. Thus we have the following class iﬁcation theorem for irreducible root

systems [ 91 , p.272]:

Figure 4.2: The exceptional Dynkin diagrams E

, E

, F

and G

Theorem 4.6.4 (Classiﬁcation of root systems). Every irreducible root system is isomor-

phic to precisely one root system from the following :

1. The classical root systems A

, n ≥ 1;

2. The classical root systems B

, n ≥ 2;

3. The classical root systems C

, n ≥ 3;

4. The classical root systems D

, n ≥ 4;

5. The exceptional root systems G

, F

, E

, and E

In the language of semisimple Lie theory (recall Chapter 3), suppose that h is a Cartan

subalgebra of a r ed u ct i ve semisimple Lie algebra g. Suppose also that ∆ is the root system

of g with respect to h. We know that ∆ is identiﬁed with the dual h

∗

by means of the

bilinear form ·, ·, and the x

(for α ∈ ∆) are the correspondi n g root vectors. Together

with a basis of the root system ∆ , the x

’s form a Chevalley basis of g. The sublattice

Q = Z∆ =





: n

∈ Z, α

∈ ∆



(4.30)

of some lattice L, generated by ∆ is the root lattice of g, and its dual

◦

= {α ∈ h : α, Q ⊆ Z} (4.31)

100

is the so called weight lattice. We now list examples of positive deﬁnite even l a tt i ces L

such that δ = L

spans h and is indecomposable. Thus, the corresponding Lie algebra g is

simple. In each case, L is generated by ∆, that is, L = Q = Z∆. The not at i ons A

, D

and E

correspond to t he standard designation s of the simple Lie algebras given above. In

each case, n = rank Q = dim h, and recall that dim g = n + |∆|.

For ℓ ≥ 1, denote by V

ℓ

an ℓ-dimensional rational vector space equipped with a positive

deﬁnite symmetric form ·, · and an orthonormal basis {v

, . . . , v

ℓ

Type A

, n ≥ 1: In V

n+1

take



n+1



i=1

: m

∈ Z,



= 0



Then, ∆ = {±(v

− v

) : 1 ≤ i < j ≤ n + 1}, |∆| = n(n + 1) and dim g = (n + 1)

− 1. As

we have mentioned above, the case A

is the case g = sl

(C).

Type D

, n ≥ 3: In V

take





i=1

: m

∈ Z,



∈ 2Z



Then, ∆ = {±v

± v

: 1 ≤ i < j ≤ n}, |∆| = 2n(n − 1) and dim g = n(2n − 1). As we

have mentioned above, the case D

is t h e same as the case A

. The sam e construct i on for

n = 2 gives A

× A

Type E

: In V

take

= Q



i=1





i=1

: either m

, . . . , m

∈ Z, or m

, . . . , m

∈ Z +

;



∈ 2Z



Then, ∆ = {±v

± v

: 1 ≤ i < j ≤ 8} ∪





i=1

: m

= ±



∈ 2Z



, |∆| = 240

and d im g = 248.

Type E

: In V

take

= Q

∩





i=1

: m

∈



∈ 2Z



101

Then, ∆ = {±v

±v

: 1 ≤ i < j ≤ 6}∪{±(v

−v

)}∪





i=1

: m

= ±(v

−v

), m



∈ 2Z



, |∆| = 126 and dim g = 133.

Type E

: In V

take

= Q

∩





i=1

: m

∈

Z, m

= m



Then, ∆ = {±v

± v

: 1 ≤ i < j ≤ 5} ∪







i=1

+ v

− v

)



: m



∈ 2Z −



, |∆| = 72 and dim g = 78.

4.7 Lie theory and Moonshine

McKay not only noticed (1.1), but also observed t hat

j(z)

1/3

= q

−1/3

(1 + 248q + 4, 124q

+ 34, 752q

+ . . .). (4.32)

The point is that 248 is the dimension of the deﬁning rep r esentation of the E

simple Lie

algebra, while 4, 124 = 3, 875 + 248 + 1 and 34, 752 = 30, 380 + 3, 875 + 2 · 248 + 1. Inci-

dentally, j

1/3

is the Hauptmodul for the genus 0 congruence group Γ(3). Thus Moonshi n e

is relat ed somehow to Lie theory.

McKay later foun d independent r el at i o n sh i p s with Lie theory [142], [17], [74], reminiscent

of his famous A-D-E correspondence with ﬁnite subgroups of SU

(C). As mentioned in

Chapter 1, M has two conjugacy classes of involutions. Let K be the small er one, called

‘2A’ in [36] (the alternati ve, class ‘2B’, has almost 100 mill i on times more elements). The

product of any two elements of K will lie in one of nine conjugacy cla sses : n a m el y, 1A, 2A,

2B, 3A, 3C, 4A, 4B, 5A, 6A, cor r esponding respectively to elements of orders 1, 2, 2, 3, 3,

4, 4, 5 , 6. It is surpr is ing that, for such a complicated group as M, that list stops at only 6

—we ca l l M a 6-transposi t i on grou p for this reason—. The punchline: McKay noticed that

those ni n e numbers ar e precisely t h e labels of the aﬃne

Dynkin diagram (see Figure

4.3). Thus we can attach a conjugacy class of M to each vertex of the

diagram. An

interpretation of t he edges in the

diagram, in ter m s of M, is unfortunately not known.

We can’t get the aﬃne

labels in a similar way, but McKay noticed that an order two

folding of aﬃ n e

gives the aﬃne

diagram, and we can ob t ai n its label s using the

Baby Monster B (the second largest sporad i c) . In parti cu l ar , let K now be the smallest

conjugacy class of involutions in B (also labeled ‘2A’ in [36]); the conjugacy classes in K

have or d er s 1, 2, 2, 3, 4 (B is a 4-transposition group), a n d these are the labels of the

aﬃne Dynkin diagram. O f course we prefer

, but perhaps that two-folding has

something to do with the fact that an or d er - two central extensi on of B is the centraliser of

an elem ent g ∈ M of order two.

102

3 2 1

2 4 3 2 1

2 4

6 5 4 3 2 1

Figure 4.3: The aﬃne Dynkin diagrams

and

with l abels.

Now, the triple-folding of the aﬃne Lie algebra

is th e aﬃne

. The Monster has three

conjugacy classes of order three. The smallest of these (‘3A’ in Atlas notation) has a cen-

traliser which is a triple cover of the Fischer group Fi’

. Taking the smalle st conjugacy

class of involutions in Fi’

, and multiplying it by it s elf, gives conjugacy classes with orders

1, 2, 3 (hence Fi’

is a 3-tran sposition group) and those not surprisingly are the labels

. Although we now understand (4.32) (see Chapter 9) and have proved the basic

Conway-Norton conjecture (see Chapter 8), McKay’s observation about

and

diagrams still have no explanati on . In [74] these patterns are extended, by relating variou s

simple groups to the

diagram with de let ed nodes.

Shortly after McKay’s E

observation, Kac [111] and James Lepowsky [132] independently

remarked that the unique level-1 highest - weight representation L(ω

) of the aﬃne Kac-

Moody algebra E

(1)

has graded dimension j(z)

1/3

. Since each homogeneous piece of any

representation L(λ) of the aﬃne Kac-Moody algebra X

(1)

ℓ

(in Kac’s notat i o n ) must carry a

representation of the associated ﬁnit e- d i m en si onal Lie group X

ℓ

(C), and the graded dime n -

sions (multiplied by an appropriate power of q) of an aﬃne algebra are modular functions

for some G ⊆ SL

(Z), this explained McKay’s E

observation. His observation (1.1) took

longer to clarify because so much of the mat h em a ti cs needed was still to be developed.

103

Chapter 5

The Moo nshi ne module

Following [67], we shall construct in this chapter a module with an associated vertex oper-

ator algebra V

♮

on which th e Monster group M acts as group of automorphis m s. We shall

call this the Monster vertex algebra or the Moonshine module. We cannot describe this

vertex algebra in detail here —in d e ed a lengthy book i s needed to do this—. However, we

shall mention some of its most basic properties.

The vertex operator algebra associated with V

♮

crowns the sequ ence of exceptional struc-

tures starting with the Golay error-correcting code and continuing with the Leech lattice.

The corresponding sequence of their automorphisms consist of the Mathi eu sporadic group

, the Conway sporadic group Co

and the Monster M.

We begi n this chapter with introductions to the two exceptional structures mentioned

above. For more det ai l s, we refer to the papers [1 30 ] - [1 31 ] , [29]-[33], to the books [86],

and to the extensive collection [39]. The history of the discoveries related to to Monstrous

Moonshine is rev i ewed in Chapter 1.

5.1 The Golay code

Let Ω a ﬁnite set with n elements. The power set ℘(Ω) = {S : S ⊆ Ω} can be viewed as an

-vector space under the operation + of symmetric diﬀerence. By a (binary linear) code

we shall understand an F

-subspace of ℘(Ω). An isomorphis m of codes is deﬁned in the

obvious way. The ca r d i n al i ty |C| of an element C of a code is called the weight of C. A

code C is said to be of type I if

n ∈ 2Z, |C| ∈ 2 Z for all C ∈ C and Ω ∈ C ,

and C is said to be of type II if

n ∈ 4Z, |C| ∈ 4 Z for all C ∈ C and Ω ∈ C .

The typ e II codes will be seen as analogues of even lattices, and will help us construct them .

For a code C , the dual code C

◦

is given by

◦

= {S ⊆ Ω : |S ∩ C| ∈ 2Z, for all C ∈ C }.

105

Thus, C

◦

is the annihi l at or of C in ℘(Ω) with respect to the natural n on s ingular symmetric

bilinear form

, S

) → |S

∩ S

| + 2Z (5.1)

on ℘(Ω) . Hence

dim

◦

= n − dim

C .

We call C self-dual if C

◦

= C , in which case n is even and dim

C =

Consider the subs p ace

E(Ω) = {S ⊆ Ω : |S| ∈ 2Z}.

The map q : E(Ω) → Z/2Z = F

given by S →

|S|

+ 2Z is a quadratic form on E(Ω), with

associated bilinear form given by (5. 1 ) . In case n ∈ 2Z, F

Ω is the radical form of q. A

subspace of a space with a quadratic form is called totally singular if the form vanishes on it.

Remark 5.1.1. In case n ∈ 4Z, the self-dual codes of type II correspon d to the (maximal )

totally singular subspaces of E(Ω)/F

Ω of dimension

− 1. Equivalently, t h e type II self-

dual codes ar e the (maximal) totally singular subspaces of E(Ω) of dimensi on

For a code C , set

w(C ) =



C∈C

|C|

∈ Z[q]

the weight distribution of C . A code with the following properties is called a Hamming

code (see [67, p.300]):

Theorem 5.1.2. There is a self-dual code of type II on an 8-element set Ω.

It can be proved t h a t the weight distribution of th e constru ct ed Hammin g cod e is 1+14q

. The Hamming code is unique up to isomor p h i sm (see, e. g. [ 13 6 ] ).

Deﬁnition 5.1.3. A code with th e following properties is called a (binary) Golay code [67,

p.301]:

Theorem 5.1.4. There is a self-dual code C of type II on a 24-element set, such t hat C

has no elements of weight 4.

It is not hard to see that the weight distribution of the constructed Golay code is 1 +

759q

+ 2, 576q

+ 759q

+ q

(see [46]). Note that 759 =









. Th e Golay code is

unique up to isomorphism (see, e. g. [136] ) . The 759 elements of the Golay code of weight

8 are called octads. We m ention some properties of the Golay code [67 , p.302].

Proposition 5.1.5. Let Ω be a 24-element set and let C be a Golay code in E(Ω).

(a) Every 5-element subset of Ω is included in a unique octad in C .

(b) Let T

be a 4-element subset of Ω. Then T

lies in exactly 5 octads. These are of the

form T

∪T

, for i = 1, . . . , 5, where Ω = ∪

i=0

is a disjoint union of 4-element sets (called

a sextet in [32]). The union of any pair of T

is an octad.

106

From thi s, it follows directly that the octads generate the Golay code. The group of

automorphisms of the Golay code C is called the Mathieu group M

= Aut C .

This is a nonabelian simple group (see Table 1.1). In fact, in the natural rep r ese ntation of

on ℘(Ω) , the complete list of submodules is

0 ⊆ Ω ⊆ C ⊆ E(Ω) ⊆ ℘(Ω).

In part i cu l a r, C /Ω and E(Ω)/C are faithful irreducible modules for M

5.2 The Leech lattice

A self-dual code C of type II based on a set Ω gives rise to an even un im odular lattice as

follows. L et

h =



k∈Ω

Fα

be a vector space with basis {α

: k ∈ Ω} and provide h t h e symmetric bilinear form ·, ·

such tha t

α

, α

ℓ

 = δ

kℓ

, for k, ℓ ∈ Ω.

For S ⊆ Ω, set α



k∈S

. Deﬁn e

Q =



k∈Ω

Zα

and for a code C based on Ω deﬁne the positive de ﬁ n i t e lattice





k∈Ω

: m

∈

Z, {k|m

∈ Z +

} ∈ C



. (5.2)

(there should be no confusion with notation in (4.24)). Then L

is even if and only if |C| ∈

4Z, for all C ∈ C . Moreover, the dual lattice L

◦

of L

(recall (4.26)) i s the correspondin g

lattice based on the dual code C

◦





k∈Ω

: m

∈

Z, {k|m

∈ Z +

} ∈ C

◦



Thus we have

Proposition 5.2.1. A code C is self-dual of type II if, and only if, the corresponding lattice

is even self-dual, or equivalently, even unimodular.

107

Now consider the following modiﬁcation of the lattice L

associated with a code C , which

we now assume contains Ω:

′



C∈C



k∈Ω

Ω

− α

)



C∈C



k,ℓ∈Ω

Z(α

− α

ℓ

) + Z(

Ω

− α

), (5.3)

where k

is a ﬁxed element of Ω. Since the lattice

∩ L

′



C∈C



k,ℓ∈Ω

Z(α

− α

ℓ

) (5.4)

has index 2 in both L

and L

′

, then L

′

is unim odular i f and only if L

is. A necessary and

suﬃcient condition for L

′

to be even is that

n = |Ω| ∈ 8(2Z + 1),

since



Ω

− α

Ω

− α



+ 1, (5.5)

for k ∈ Ω, and we ﬁnd

Proposition 5.2.2. If n ∈ 8(2Z + 1) and the code C is self-dual of type II, then the

corresponding lattice L

′

is even unimodular.

Deﬁnition 5.2.3. The Leech lattice is the even unim odular lattice

Λ = L

′

for th e case n = 24 and C th e Golay code (recall Theorem 5.1.4).

The lattice Λ has no ‘short’ elements, i. e., Λ

= Ø (using notation (4.24)). This can be

checked from equation (5.3) and the correspon d i n g property of the Golay code, but we

shall prove Λ

= Ø and recon st r u ct Λ inst ead by using another principle, which will be an

analogue for latt i ces of Remark 5.1.1 for codes.

Let L be an even uni m odular latti ce of rank n and with form · , ·. For our speci al purpose,

now we provide L with the following rescaled form:

α, β

1/2

α, β for α, β ∈ L (5.6)

and by abuse of notation we drop the subscript 1/2. With respect to the new form ·, ·, L

has th e following properties:

• α, α ∈ Z, for all α ∈ L.

•



det[α

, α

]

i,j



, for a base {α

, . . . , α

} of L.

108

As we did in Section 4.4 (bu t keeping in mind the rescaled no rm ), set

L = L/2L,

an n-dimensional vector space over F

, and write α → ˘α for the canonical m ap. Since the

original lattice L is even, the map q

L → Z/2Z = F

given by

˘α → α, α + 2Z

deﬁnes a quadratic form on

L with associated biline ar form c

L ×

L → F

determined by

(˘α,

β) → 2 α, β + 2Z.

These forms are nonsingul ar since the original l at t ic e L is unimodular. From these deﬁni-

tions we have

Proposition 5.2.4. Let M be a lattice such that 2L ⊆ M ⊆ L. Then, M is even unimod-

ular with respect to the new form ·, · if, an d only if,

M = M/2L is a (maximal) totally

singular subspace of

L of dimension

We shall apply this last principle to the direct sum of three copies of the r oot lattice Q

(recall Section 4. 6 ) which is an even unimodular lattice of rank 8. For brevity, set

Γ = Q

and provide Γ with the rescaled form (5.6) as above. Using the Hamming code (Theorem

5.1.2), we shall ﬁrst show that

Γ contains complement ar y 4-dimensional totally singular

subspaces.

We can describe the lattice Γ as follows:

Γ =



k,ℓ∈Ω

ℓ

) + Z

Ω





k∈Ω

: either m

, . . . , m

∈

Z, or m

, . . . , m

∈

Z +

;



∈ Z



Now we identify Ω with the projective line over the 7-element ﬁeld: Ω = P

) = F

∪{∞}.

Consider the sets of squares and non-squares

Q = {x

: x ∈ F

} = {0, 1, 2, 4},

N = Ω − Q = {3, 5, 6, ∞};

and d eﬁ n e subspaces

= N + i|i ∈ F

,

= −N − i|i ∈ F

,

109

of E(Ω). Then it follows that C

and C

satisfy the prop er t i es of Theorem 5.1.2. Now,

consider the latti ces

Φ = L

for C = C

Ψ = L

′

for C = C

Then, Φ and Ψ are even unimodular lattices by Propositions (5.2.1) and (5.2.2). Moreover,

2Γ ⊆ Φ, Ψ ⊆ Γ, and since C

+ C

= E(Ω), we see that

Φ + Ψ = Γ.

Proposition (5.2.4) thus gives

Proposition 5.2.5. We have a decomposition

Γ =

Φ ⊕

Ψ (5.7)

into 4-dimensional totally singular subspaces. In particular, Φ + Ψ = Γ and Φ ∩ Ψ = 2Γ.

Using the decomposition (5.7) we shall now reconstruct the L e ech lat t i ce by analogy with

the con st r u ct i on of the Golay code in Theorem 5.1.4. Set

= Γ ⊕ Γ ⊕ Γ

the orthogonal direct sum of three copies of Γ, equipped with the modiﬁed form (5.6). In

, set

Λ = {(φ, φ, 0) : φ ∈ Φ} ⊕ {(φ, 0, φ) : φ ∈ Φ} ⊕ {( ψ, ψ, ψ) : ψ ∈ Ψ}. (5.8)

Then we have [67, p.306]

Theorem 5.2.6. The lattice Λ in (5.8) is an even unimodular latti ce such that Λ

= Ø.

Moreover, it coincides with the Leech lattice in Deﬁnition 5.2.3.

The Leech lattice is the uniqu e positive deﬁnite even unimod u l ar lattice Λ of rank 24 with

= Ø, up to isometry (see [153], [30]), and the E

-root lattice is the unique positive

deﬁnite even unimodula r lattice of rank 8 up to isometry. Thus the lattices Γ, Φ and Ψ

with its or i g inal bil inear form, are all isometric. In fact, the construction (5.8) expresses

the Leech lattice as the non-orthogonal direct sum or t h r ee rescaled copies of the E

-root

lattice. Sometimes the L eech lattice Λ is denoted as Λ

The grou p of isometries of the Leech latti ce is called the Conway group Co

(or ·

)

= Aut(Λ, ·, ·)

= {g ∈ Aut Λ : gα, gβ = α, β for all α, β ∈ Λ}.

Its quot i ent by the central subgroup ±1 is called the Conway group Co

= Co

/±1.

We cite some basic facts about this gr oups [31, 32]:

110

• Co

equals i t s commutator subgroup: Co

= [Co

, Co

];

• Cent Co

= ±1;

• Co

is a nonabeli an simple group (see Table 1.1);

• Co

acts fait h fu l l y and irreducibly on λ/2Λ.

Only as a comment, we have constructed t h e L ee ch lattice as the lattice L

′

based on the

Golay code. The lattice L

based on the Golay code is also even unimodular, but (L

)

= Ø.

In fact, (L

)

= {±α

: i ∈ Ω}. This lattice L

is called the Niemeier lattice of type A

and is somet im es written as L

= N(A

). We have encountered a t h i r d even unimod u l ar

lattice of r an k 24, namely, Γ

. Altogether, there are 24 even unimodular lattices of rank

24, u p to isometry, called the Niemeier latti ces (see [153]).

Next, we shall describe and co u nt th e shortest nonzero elem ents of the Leech lattice. In

the notation of the beginning of this section, for S ⊆ Ω let ǫ

be the involution of h given

: α

→



−α

if k ∈ S

if k /∈ S

for k ∈ Ω. It follows from relations (5.2), (5.3), (5.4) and (5.5) that Λ

is composed of three

types of elements:

= Λ

∪ Λ

where

= {

: C ∈ C , |C| = 8, S ⊆ C, |S| ∈ 2Z};

= {±α

± α

ℓ

: k, ℓ ∈ Ω, k = ℓ};

= {ǫ

(

Ω

− α

) : C ∈ C , k ∈ Ω}.

Counting, we ﬁnd that

|Λ

| = |Λ

| + |Λ

| (5.9)

= 759 · 2





· 2

+ 24 · 2

(5.10)

= 196, 560. (5.11)

5.3 The Monster vertex algebra V

♮

and the Griess al-

gebra B

Now that we have the Leech lattice available, we can construct one of our main objects of

study. We have alread y s een t hat t he Leech lattice i s t h e unique even unimodular lattice

111

of rank 24, such that has no elements of norm 2.

First, using the Leech lattice we form the untwisted space

= S(

−

) ⊗

C[Λ] (5.12)

as in (4.20). Let

L be a central extension of a l at t ic e L by a ﬁnite cyclic group κ =

{k|k

= 1} of order s, and den ot e by

: L × L → Z/sZ

the associated commutator map, so that

aba

−1

= κ

(¯a,

, for a, b ∈

We make the following special choices: we ﬁx the central extension

1 −→ κ −→

Λ −→ Λ −→ 1, (5.13)

where κ

= 1 , κ = 1 (i. e., s = 2), and where the com mutator map is the alternating

Z-bilinear map

(α, β) =  α, β + 2Z, for α, β ∈ Λ. (5.14)

Then, t aki ng a 2-th primitive root of u nity ξ = −1, we have

C[Λ] = C[

Λ]/(κ + 1)C[

Λ]

and

c(α, β) = ( − 1)

α,β

, for α, β ∈ Λ,

where c : Λ × Λ → C

is the map (α, β) → ξ

(α,β)

. So that, as operators on V

, κ = −1

and

ab = (−1)

¯a,

b

ba, for a, b ∈

Λ.

The automorphisms of

Λ which induce the involution −1 on Λ are automatically involutions

and are parametrized by t h e quadratic forms on Λ/2Λ with associated form induced by

(5.14). Among these, we ﬁx the distinguished involution θ

determined by the canonical

quadratic form q

given in (4.28)

: Λ/2Λ → Z/2Z, with α + 2Λ →

α, α + 2Z.

Then, we have an involution θ

Λ →

Λ given by

a → a

−1

¯a,¯a/2

Besides the general properties θ

= 1 and θ

) = a

−2

, for all a ∈

Λ, we observe that

(a) = a

−1

if ¯a ∈ Λ

112

We also form the twisted space

= S(

−

) ⊗

Z+

where T is any Λ-module such that κ · v = ξv, for all v ∈ T . Keepin g in mind the choices

above, we set

K = {θ

(a)a

−1

: a ∈

Λ} = {a

¯a,¯a/2

: a ∈

Λ},

which is a central subgroup of

Λ such that K = 2Λ. Then,

Λ/K is a ﬁnite group which is

a central extension

1 −→ κ −→

Λ/K −→ Λ/2Λ −→ 1,

with commutator map indu ced by (5.14) and with squaring m ap the quadratic form q

Since Λ is unimodular, q

is non - si n gular, and

Λ/K i s an extraspecial 2-group with

Λ/K| = 2

. (5.15)

In fact, in the twisted space V

we take T to be the can o n i cal

Λ-module described in (5.12).

Of cour se for a ∈

Λ, we have θ

(a) = a as operators on T .

Now we can deﬁne the Moonsh ine module —the space on which the Monster group will

act—. Recal l that θ

acts in a natural way on V

given by

: x ⊗ i(a) → θ

(x) ⊗ i( θ

(a)), for x ∈ S(

−

) and a ∈

Λ; (5.16)

and on V

, by

: x ⊗ τ → θ

(x) ⊗ ( −τ ) = −θ

(x) ⊗ τ, for x ∈ S(

−

) and τ ∈ T. (5.17)

Let V

and (V

)

be the subspaces of V

and V

of θ

-invariant elements. We know

that for v ∈ V

the component operators of both the untwisted and the twisted vertex

operators Y (v, z) preserve the respective ﬁxed spaces V

and (V

)

Deﬁnition 5.3.1. We deﬁne the Moonshine module to be the space

♮

= V

⊕ (V

)

The symb o l ♮ is for ‘natural’. For v ∈ V

, we form the vertex operator

Y (v, z) = Y

(v, z) ⊕ Y

Z+

(v, z)

acting on the larger space

= V

⊕ V

. (5.18)

Similarly, for the component operators of Y (v, z) we write v

= v

⊕ v

and x

(n) =

(n) ⊕ x

(n) on W

, for v ∈ V

, n ∈ Q. Then

· V

♮

⊆ V

♮

, x

(n) · V

♮

⊆ V

♮

113

if v ∈ V

. In fact, V

♮

can be given the structure of a vertex operator algebra, the Monster

vertex algebra. V

♮

has a conformal vector ω of central charge 24. Thu s, the linear maps

: V

♮

→ V

♮

given by L

= ω

i+1

satisfy

, L

] = (i − j)L

i+j

+ 12



i + 1



i+j,0

♮

and thu s give a representation of the Virasoro algebra in which the central element c is

represented by 24 · 1

♮

(recall (4.13)).

Because dim h = 24 and Λ

= Ø, the space V

♮

has some special structu ra l features. First,

♮

is integrally graded, with degrees bounded below by − 1:

♮



n∈Z

♮

with V

♮

= 0, for all n < −1. In particular,

♮

−1

= F

ℓ

(1),

♮

= 0,

♮

= f ⊕ p,

where, i n more detail

f = S

(h) ⊕



α∈

p = h ⊗ T.

Deﬁnition 5.3.2. We deﬁne the Griess module to be the space

B = V

♮

= f ⊕ p.

We have counted the elements of Λ

in (5. 11 ) , and we ﬁnd that

dim V

♮

−1

= 1,

dim V

♮

= 0,

dim B = dim V

♮

= 196, 884;

since

dim f = 300 +

(196, 560) = 300 + 98, 280 = 98, 580 (5.19)

and

dim p = 24 · 2

= 98, 304. (5.20)

114

In fact we have (recall Chapter 1)



n∈Z

(dim V

= J(z) = q

−1

+ 0 + 196, 884q + 21, 493, 760q

+ . . . ,

where q = e

2πiz

, z ∈ H.

It is shown in the last four chapters of [67] that the Monster group M acts as a group

of automorphisms of V

♮

. The subgroup of M preserving the subspaces V

and (V

)

the centralizer of an involution in M. This is an exten s io n of th e extraspecial group

Λ/K

of order 2

in (5.15) by Conway’s sporadic group Co

, which we have seen is re la t ed to

the Leech lattice Λ. A crucial part of the Frenkel-Lepowsky-Meurman constructi o n is to

ﬁnd an involution in the Monster M which acts on V

♮

, but which does not preserve the

subspaces V

and (V

)

The confor m al vector ω ∈ V

♮

lies in a 1-dimensional subspace Cω ⊆ V

♮

, invariant under

the action of M. The co m p l em entary submodule of Cω in V

♮

gives the smallest non-

trivial representation of M of degree 196,883. This then, is the explanation of McKay’s

observation (1.1) con s idered Chapter 1. The Monster vertex algebra V

♮

is a graded module

whose graded components have dimension given by th e coeﬃcients of the J(z) function,

and the Monster M acts on each graded component. On V

♮

, the Jacobi identity takes the

following simple form for vertex operators parametrized by V

[67, p .3 16] :

Theorem 5.3.3. For v ∈ V

, we have

Y (v, z) =



n∈Z

−n−1

on V

♮

that is, Y (v, z) involves only integral powers of z. For u, v ∈ V

, we have

[Y (u, z

) ×

Y (v, z

)] = z

−1



− z



Y (u, z

)Y (v, z

) − z

−1



− z

−z



Y (v, z

)Y (u, z

)

= z

−1



− z





Y (u, z

)v, z



on V

♮

. In particular, V

is a vertex operator algebra of central charge 24 and (V

)

is a

-module.

Moreover, we know the following [ 67 , p.317]:

Theorem 5.3.4. The space f is a commutative nonassoc iati ve algebra with identity under

the product

u × v = u

· v,

and the bilinear form

u, v = u

· v

115

is nonsingular, symmetric and associative. The space V

♮

is a graded module for the com-

mutative aﬃnization

f of f under the action π :

f → End V

♮

deﬁned by

π :



u ⊗ t

→ x

(n) for u ∈ f, n ∈ Z

e → 1.

(5.21)

One of the main results obtained in the work of Frenkel, L epowsky and Meurman [67] is

an extension of this action to a rep r esentation

B → End V

♮

(5.22)

of a la r ger commutative aﬃ n i zat i on by cross-bracket on V

♮

, where the space B is given the

structure of a commutative nonassoci a ti ve algebra with identity, and with a nonsingular

symmetric associative form in the fol l owing way: the product × an d the form ·, · on B

extend those on f. For u ∈ f and v ∈ p, we use the product in Theorem 5.3.4 and the

commutativity of this product on f as mot i vation to deﬁne

u × v = v × u = u

· v.

Similarly, we use the bili n e ar form in Theorem 5.3.4 and the symmetry of this form on f as

motivation to d eﬁne

u, v = v, u = u

· v = 0

(the fact that u

·v = 0 is obtained by consideration of the gradation of (V

)

). Now, the

identity element

ω on f is also an identity element on B. We deﬁ n e next a nonsingular

symmetric bilinear form  ·, · on p = h ⊗ T by the formula

h

⊗ τ

, h

⊗ τ

 =

h

, h

τ

, τ

 ( 5. 23)

for h

∈ h, τ

∈ T . Finall y, we deﬁne a product × on p so that p × p ⊆ f, and uniquely

determined by the nonsingul a r ity of the form on f and the associativity condition

u, v × w = u × v, w, for u, v ∈ p, w ∈ f. (5.24)

The commutativity of this product on p follows from the explicit formula for it given below.

Deﬁnition 5.3.5. The resulting nonassociative alg eb r a B equipped with this form ·, · is

called the Griess algebra.

Here, the Griess algebra B = f ⊕p is actually a slight modiﬁcation, with a natural identity

element, of the algebra deﬁned in [79], [80]. Summarizing,

Proposition 5.3.6. The Griess algebra B is a commutative nonassociative algebra with

identity element

ω ∈ f, and the form ·, · on B is nonsingular, symmetric and associative.

We have

f × f ⊆ f,

f × p ⊆ p,

p × p ⊆ f,

116

with explicit formulas given by:

× h

= 4g, h gh,

× x

= g, ¯a

× x







0 if ¯a,

b = 0, ±1

if ¯a,

b = −2

¯a

if ab = 1

for g, h ∈ h, a, b ∈

and x

= i(a) + i(θa) = i(a) + θi(a);

)

· (h ⊗ τ) =

(h − 2¯a, h¯a) ⊗ a · τ,

)

· (h ⊗ τ) = (g, h +

g, gh) ⊗ τ,

for g, h ∈ h, a ∈

, τ ∈ T ;

⊗ τ

) × (h

⊗ τ

) =



+ h

, h





τ

, τ

 ⊗ a · τ +

128



a∈



h

, h

 − 2¯a, h

¯a, h





τ

, a · τ

x

for h

∈ h, τ

∈ T . We also have

f, p = 0,

and explicit formulas for the form on f and p given by:

g

, h

 = 2g, h

g

, x

 = 0,

x

, x

 =



0 if ¯a = ±

2 if ab = 1

for g, h, a, b as above and (5.23)

h

⊗ τ

, h

⊗ τ

 =

h

, h

τ

, τ

.

The identity element satisﬁes 

ω,

ω = 3.

The signiﬁcance of B is that Griess, who introduced thi s al gebra, constructed a group

of automorphisms of it, preserving the form ·, ·, and showed this group to be a ﬁnite

simple group: the Monster M (see [80]). In fact, Tits showed that the Monster is the full

automorphism group of B (see [174], [175]). Having reconstr u ct ed the Griess algebra using

properties of vertex operators, we shall also reconstruct the Mons t er using properties of

vertex operators, and exh i b i t a natural action of it on V

♮

117

5.4 The construction of M

We now sketch the discoveries and con st r u c ti o n s of some of the generation t h r ee of the

Happy Family sporadic simple groups and the pariahs, and make some comments on their

properties. The ﬁrst generation is related to the Golay code C and the Mathieu ﬁve spo-

radic groups. The second one, is related to the Leech lattice and the Conway sporadic

groups. For a detailed discussion on thi s two other generations of the Happy Family see

the book of Griess [86].

The third generation of the Happy Family [80] consist of eight simple groups (see Tabl e 5 .1 )

which are involved in the largest sporadic simple group, the Monster M. The constru c ti o n

of M depends on constructio n and analysis of a certain commutative nonassociative algebra

B, of dimension 196,884, over the rational ﬁeld Q. This is the Griess algebra we alread y

have st u d i ed in Section 5.3 . This algeb r a B plays the role of the Golay code in the First

generation, and the Leech lattice in the second.

Group Discoverer (d a t e) First con st r u c t io n (date)

He D. Held (1968) G. High m an, J. McKay (1968?)

B. Fischer (1968) B. Fischer (1969)

′

B. Fischer (1968) B. Fischer (1969)

HN K. Harad a (1973) S. Nor to n (1974)

Th J. Thompson (1973) P. Smith (1974)

B B. Fischer (1973) J. Leon , C. Sims (1977)

M B. Fischer, R. Griess (1973) R. Gri ess (January, 1980)

Table 5.1: Third generation of the Happy Family.

In a sequence of steps, we now proceed to deﬁne and establi sh the basic properties of a

group C which will act naturally on V

♮

, B and

B, and which moreover will act compatibly

with the appropriate vertex operators. This group will be the centralizer of an involution

in th e Monster.

Starting with t h e central extension

Λ (5.13 ) we ﬁrst set

= {g ∈ Aut

Λ : ¯g ∈ Co

}, (5.25)

where ¯g is the automorphi sm of the Leech lattice Λ induced by g and Co

is the isometry

group of Λ (5.9). We know that gκ = κ automatically. We have also mentioned that the

sequence

1 −→ Hom(Λ , Z/2Z)

∗

−→ C

−

−→ Co

−→ 1 (5.26)

118

is exact, where the pullback λ

∗

Λ →

Λ is given by

a → aκ

λa

for λ ∈ Hom(Λ, Z/2Z). Moreover, we have natural identiﬁcations

Hom(Λ, Z/2Z) = Λ/2Λ = Inn

Λ,

so that the exact sequence (5.26) can be writt en

1 −→ Inn(

Λ) ֒→ C

∗

−→ Co

−→ 1. (5.27)

Now, C

induces a group of automorphisms of the extraspecial g r ou p

Λ/K since C

preserves

K, and we have a natural isomorphism ϕ : C

→ Aut(

Λ/K). We claim that

Ker ϕ = θ

.

In fact, it is clear that θ

∈ Ker ϕ. On the other hand, Ker ϕ ∩Inn

Λ = 1, since K ∩κ = 1.

Hence, Ker ϕ is isomorph i c to it s image in Co

by (5.27). But,

Ker ϕ acts triviall y on λ/2Λ,

so that by the faithfulness of the action of Co

on Λ/2Λ, we have

Ker ϕ ⊆ ±1, p r oving

the clai m .

Set C

= ϕ(C

) ⊆ Aut(

(Λ)/K). Then we have an exact sequence

1 −→ Inn(

Λ) −→ C

∗

−→ Co

−→ 1. (5.28)

Here Co

acts in the natural way on Inn

Λ = Λ/2Λ, and it follows from the properties of

the Conway group Co

that C

equals i t s commutator subgroup and has trivial center:

• C

= [C

, C

];

• Cent C

= 1.

We recall (see [67]) that the extraspecial group

Λ/K, gives the exact sequence

1 −→ F

−→ N

Aut T

(π(

Λ/K))

int

−→ Aut(

Λ/K) −→ 1, (5.29)

where π denotes the faithful representation of

Λ/K on T , N

Aut T

(π(

Λ/K)) is the normalizer

of π(

Λ/K) on Aut T , and

int(g)(x) = gxg

−1

for g ∈ Aut T , x ∈

Λ/K = π(

Λ/K) (T the Λ-module used in the twisted space V

). Set

∗

= {g ∈ N

Aut T

(π(

Λ/K)) : int(g) ∈ C

119

so that we have the commutative d i agr a m with exact rows

∗



Aut T

(π(

Λ/K))

Aut(

Λ/K)

Also set C

= [C

∗

, C

∗

]. We shall now show that C

contains − 1 and in fact all of π(

Λ/K).

Since

int(π(

Λ/K)) = Λ/2Λ = Inn

Λ, (5.30)

we see that π(

Λ/K) ⊆ C

∗

, ans so −1 = π(κK) ∈ [π(

Λ/K), π(

Λ/K))] ⊆ C

. But since Co

acts irr e d u ci b l y on Λ/2Λ, we have Inn

Λ = [Inn

Λ, C

], and it follows that

π(

Λ/K)) = [π(

Λ/K)), C

∗

] ⊆ C

. (5.31)

We claim that the sequence

1 −→ ±1 ֒→ C

int

−→ C

−→ 1 (5.32)

is exact. By the proper t ie s of C

, all we need to show it that C

∩F

= ±1 . To see this,

we use the fact that the

Λ/K-module T has a Q-for m ,

∼

Ind

Λ/K

Φ/K

⊗

where ψ

is any rational-valued character of

Φ/K such that ψ

(κK) = −1, and this gives

us a

Λ-invariant Q-subspace T

of T such that the canonical map

⊗

F → T

is an isomorphism . Let C

∗,Q

= C

∗

∩ Aut T

. We have an exact sequence

1 −→ Q

−→ C

∗,Q

−→ C

−→ 1,

and so C

∗

= C

∗,Q

, and C

⊆ C

∗,Q

, implying that C

∩F

⊆ C

∗,Q

∩F

= Q

. But since

det C

= 1, we also have C

∩ F

⊆ {µ ∈ F

: µ

= 1}, proving the claim.

Using ( 5. 28 ) we have a ma p

int =

−

◦ int : C

→ Co

and by (5.30) and (5.31), th en π(

Λ/K) ⊆ Ker int. Consideration on the order of C

shows

that t h e sequence

1 −→

Λ/K

−→ C

int

−→ Co

−→ 1 (5.33)

120

is exact. Summari zing, we have an extension C

of Co

by Λ/2Λ (5.27), an extension C

of Co

by Λ/2Λ (5.28), and an extension C

of Co

by the extraspecial group

Λ/K (5.33).

Now, form the pullback

C = {(g, g

) ∈ C

× C

: ϕ(g) = int(g

)},

so that we have the commutative d i agr a m of surjections



int

Set

= (θ

, 1),

θ = (1, −1) ∈

C. Then

Ker π

= 

θ,

Ker π

= 

,

and

Cent

C = 

 × 

θ = Ker(ϕ ◦ π

since Cent C

= 1 and θ

∈ Cent C

from the deﬁnition.

We are ﬁnally ready to deﬁne the gro u p C: Set

C =

C/

θ. (5.34)

Then, t h e diagram above enlarges to the commutative diagram of surjection s



int

Also, Ker π

= 

θ an d

Cent C = z|z

= 1 = Ker σ,

where

z = π

(

) = π

(

θ) (5.35)

(no confusion should arise between this notation and our formal variable notation). We

have the exact sequence

1 −→ z ֒→ C

−→ C

−→ 1.

121

Proceeding as in (5.33), we have a map ¯σ =

−

◦ σ : C → Co

from (5. 28 ) . Moreover, there

is a canonical embedding ν :

Λ/K →

C given by

gK → (int(g), π(g)),

since ϕ( i nt(g)) = int(π(g) ) . The result is an exact sequen ce

1 −→

Λ/K

◦ν

−→ C

¯σ

−→ Co

−→ 1,

and we have proved

Proposition 5.4.1. The group C is an extension of Co

by the extraspecial group

Λ/K.

The nontrivial central element of

Λ/K identiﬁes with the nontrivial central element of C.

Now that the grou p C is constructed we shall set up its canonical acti on on the Moonsh i n e

module V

♮

. Fi r st , we shall deﬁne an action of th e larger group

C on t h e larger space

(see (5.18)). For g ∈ Co

and Z = Z or Z +

, let g also denote the unique algebra

automorphism

g : S(

−

) → S(

−

(where S(

−

) is an abbreviation for S(

−

) ⊗

T , for the appropr i at e m odule T ), such that

g agrees with its na t u r al action on

−

. For g ∈ C

, let g a lso denote the operator

g : F[Λ] → F[Λ] given by i(a) → i(ga),

for a ∈

Λ; note that thi s is well deﬁ n e d sin ce gκ = κ. For k = (g, g

) ∈

C, let k also denote

the operator

k = ¯g ⊗ g ⊕ ¯g ⊗ g

on W

= S(

−

) ⊗

F[Λ] ⊕ S(

−

) ⊗

Z+

T . This clearly gives a faithful representation of

on W

. In fact, this representation is even faithful on a small subspace of W

, for i n st ance

T ⊕ p.

The action of

C on W

extends the action of θ

similar to that d eﬁned in (5.16) and (5.17)

for the case W

, in such a way t hat this operator corresponds to the elem ent

θ = (θ

, −1)

C. We have

C · V

♮

⊆ V

♮

From the deﬁnitions of C and V

♮

and the last paragraph, we see that C acts i n a natur a l

way on V

♮

: for k = (g, g

) ∈

C, π

(k) acts as the operator

(k) = ¯g ⊗ g ⊕ ¯g ⊗ g

. (5.36)

This action of C on V

♮

is faithful, even on p = h⊗T . The decomposition V

♮

= V

⊕( V

)

in Deﬁnition 5.3.1 is t h e eigenspace decomposition with respect to the central involuti on z

in C (5.35), and we introduce corresponding notation

= V

= {v ∈ V

♮

: z · v = v},

)

= V

−z

= {v ∈ V

♮

: z · v = −v}.

122

Note that V

♮

= V

⊕V

−z

and C · V

⊆ V

, C · V

−z

⊆ V

−z

.The actions of C on V

♮

and of

C on W

preserve t h e homogeneou s subspaces with respect to the gradings. The group C

acts on the algebra B = V

♮

(Deﬁnition 5.3.2) and in fact preserves the summands f and p.

From the deﬁniti on of the product × and the form ·, · on B (see Proposition 5.3.6), we

ﬁnd [6 7 , p.328]:

Proposition 5.4.2. The group C ac ts faithfully as automorphisms of the Griess algebra

B and as isometries of ·, ·.

In fact, in checking that the form (5.23) on p is preser ved by C, we use the fact that the

form  ·, · on T is C

-invariant:

gτ

, gτ

 = τ

, τ

, for all g ∈ C

, τ

∈ T.

Recall from Theorem 5.3.4 that V

♮

is a gra d ed module for the commutative aﬃnization

of f. We shall relate this structure to the action of C. Given a commutative nonassociati ve

algebra b with a symmet r ic form and given a group G of linear automorphisms of b, we let

G act as linear automorphisms of

b by

g · e = e,

g · (u ⊗ t

) = (g · u) ⊗ t

for g ∈ G, u ∈ b and n ∈ Z. If G acts as algebra automorphisms and isometries of b, then

G acts as algebra automorphism s of

b. Suppose now that V is a graded

b-module and that

G acts as linear automorphisms of V , preserving each homogeneous subspace V

. Then,

we call V a graded (G,

b)-module if

gxg

−1

= g · x as operators on V,

for g ∈ G, x ∈

b. By Propositi on 5.4.2 we have

Proposition 5.4.3. The Moonshine module V

♮

is a graded (C,

f)-module, and C acts as

automorphisms of

f, and in fact of

5.5 Improvements on the Construction

We have already seen in (5.36) and Proposition 5.4.2 that the group C acts as automor-

phisms on these main structur es: the Griess al ge b r a B and the Moonshine module V

♮

. In

fact, th e idea for complete the const r u c ti o n of the Monster group M lies in enlarge C to a

group M (the Monster) of aut o m or p hisms and isometries of B (and hence automorphisms

B), and make V

♮

into a ( M,

B)-module. Furt h er m or e , we should also deﬁne vertex op-

erators Y (u, z) on V

♮

, for all v ∈ V

♮

in order to extend an action of C on V

♮

(consequence

of the action of

C on W

) of type

kY (v, z)k

−1

= Y (kv, z), for all k ∈ C, v ∈ V

123

to all M and V

♮

. We will not describe the complete construction of M (this requi r es in fact

a complete book). For details on the complete construction, the reader may consult the

paper of Griess [80], or the last three chapters of [67]. Although, we mention some notably

properties.

Under t h e action of C, B breaks into the following invariant subspaces:

Fω, {u ∈ S

(h) : u, ω = 0},



a∈

, h ⊗ T (5.37)

of dim en si on s 1, 299, 98280 and 98304, r espectively. Of cou r se, C in fact ﬁxes ω:

C · ω = ω.

It can be shown that each of the invariants subspaces is absolutely irreducible under C; for

instance, h ⊗ T is irreducible since T is irreducible under t h e extraspeci al group

Λ/K and

h is irreducible under Co

. Before the Monster was proved to exist, it was pos t u l at ed to be

a ﬁni te simple group containing the group C as the centralizer of t h e involutio n z ∈ C and

it was believed to have a 196,883-dimensional irreducible modul e con si st i n g of the direct

sum of the las three C- m odules listed in (5.37), or rather, abstract C-modules isomorphic

to them. B y 1976, Norton proved the exist ence of an i nvariant commutative nonassociative

algebra and nonsingular associative symmetric bilinear form on this module if it and the

Monster existed (c. f. [ 80 ] ) , though his met h od gave no descrip t i on (he also made assump-

tions on the conjugacy classes). By constraining th e possibilities for such an algebr a and

form on the direct sum of the C-modules, Griess was able to determine an algebra and a

form admitting an automorphism outside th e group C. The group generated by C and this

automorphism had the requ i re d proper t i es.

The origi n al deﬁniti on of the algebra structure in [80] was complicated, due mainly to sign

problems. The existence o f the mentioned i r r ed ucible representation of the Monster of

degree 196,883 was predicted in 1974 by Griess [78]. With the Norton’s improvement of

the existence for such a nontrivial commutative non associative algebra struct ure o n this

module, an d the st u d y of the automorphisms outside of C, by surveying the actions of

many subgroups of C on the space B, the result was the system of structure con st ants in

[80, Table 6.1], and the formula for an extra automorphism in [80, Table 10.2]. Prove that

the linear automorphism so deﬁned preserves th e algebra structure was the hardest part o f

the con st r u ct i on in [80].

Many improvements on [80] were made by Tits [174, 175, 176], who showed that some

deﬁnitions of [80] based on guesswork may be based on a more thorough analysis. A new

style construction was made by Conway in [34], [39], using a Moufang loop to ﬁnesse the

sign prob l em s so prominent in th e original version. This loop (a nonassociati ve group)

has order 2

and is a kind of 2-cover of the binary Golay code; its creation was an idea

124

of Richard Parker (see [81]), which deﬁnes and gives the foundat i on of th e class of loops

called code loops. This theory is usually developed in the theory of p-locals in sporadic

groups and Lie groups [81, 83, 84, 85].

The algebra B is not a classic nonassociative algebra. An algebra of dimension n satisﬁes

a nontrivial polynomial identity of degree at most n + 1; B satisﬁes no nontrivial identity

in co m muting varia b l es of degree less than 6 [82]. In [80], the subgroup of Aut B generated

by C and the particular extra aut om or phism was identiﬁed as a simp le group of the right

order, thus proving t h e existence of a simple group of the right order and local properties.

The full automorph is m group of a ﬁni t e dimensional algebra is an algebraic group. In [175],

Tits showed th a t Aut B was exact l y the Monster M. In [34], Conway gave a short argument

with idempotents in B that Aut B i s ﬁnit e and in [175], Tits i d entiﬁed the centralizer of

an involution in Aut B as C (not a larger group). The proof that the group order is right

involves quoting harder theorems (see chapter 13 of [80]). In 1988, a uniqueness proof for

the Mon st er was given by Griess, Meierfrankenfeld and Segev [87].

We close our discussion with this uniqueness result. First, we deﬁne a grou p of Monster-type

to be a ﬁnite group G containing a pai r of involutions z, t s u ch that

• C(z)

∼

1+24

;

• C(t) is a double cover of Fischer’s {3, 4}-transposition group (discovered by Fischer

in 1973, and later called the Baby Monster B).

Is follows that such a grou p is simple. See [87] for a fuller discussion of the hypotheses.

Theorem 5.5.1 (Uniqueness of the Monster). A group of Monster-type i s unique up to

isomorphism.

125

Chapter 6

Intermezzo: the j functio n

The j function plays an im portant role in modern number theor y. It classiﬁ es the family

of ellipti c curves over the complex ﬁeld C. In fact, the j functions serves as the moduli

space of this family of cur ves. In this and next chapters, our purpose is to detail in a better

form some aspects of the ori gi n a l Conway-Norton conjecture. First, we will give here some

background theory of modular forms and Hecke operators, and subsequently we ou t li n e

some featu r es of replicable functions introduced in [38].

6.1 The mod ul ar gro up

Let H denote the upper half-plane of C we have introduced in Chapter 1. That is H =

{z ∈ C : Im(z) > 0}. Let SL

(R) be the group of matrices (

a b

c d

) with real entries, such

that ad − bc = 1. We made SL

(R) act on C by: if G = (

a b

c d

), th en

Gz =



a b

c d



· z =

az + b

cz + d

One can easily checks the formula

Im(Gz) =

Im(z)

|cz + d|

from which can be showed that H is stable under the action of SL

(R). We have also

seen in Chapter 1 that the element −I =



−1 0

0 −1



of SL

(R) acts trivially on H. Thus

we can consider the grou p PSL

(R) = SL

(R)/{±I} which operates —and in fact, acts

faithfully—, and one can even show that it is the group of all analytic automo r p h i sm s of

Let SL

(Z) be the subgroup of SL

(R) consisting of the matrices with integer entries. It

is a discrete subgroup of SL

(R), t hus it acts discontinually on H.

Deﬁnition 6.1.1. The group Γ = P SL

(Z) = SL

(Z)/{±I} is c al le d the modular group.

It is the image of SL

(Z) on P SL

(R). We denote by H/Γ the set of action orbits of Γ on

For simplicity, if G is an element of SL

(Z), we will use the same symbol to denote it s

image in the modular gr oup Γ. Let S = (

0 1

−1 0

) and T = (

1 1

0 1

) denote the el em ents of Γ

127

introduced in Chapter 1. We have seen that

Sz = −

, T z = z + 1,

= 1, (ST )

= 1.

On the other hand, let D the subset of H formed of all points z such that |z| ≥ 1 and

|Im(z)| ≤ 1 /2. The Figure 6.1 below represents the transforms of D by some of the

elements of the group Γ.

−1

−1/2

1/2

τ + 1

−1/σ

−1

S T S

ST S ST

−1

T ST

Figure 6. 1: Fun d am ental domain D and some of its images by S and T .

We will show that D is a fundamental domain for the action of Γ on the half-plane H.

More pr eci sel y :

Theorem 6.1.2. 1. For every z ∈ H there exists G ∈ Γ such that Gz ∈ D.

2. Suppose that t wo distinct points z, z

′

of D are congruent modulo Γ. Then, Re(z) = ±

and z

′

= z ± 1, or |z| = 1 and z

′

= −

3. Let z ∈ D and let I(z) = {G ∈ Γ : Gz = z} be the stabilizer of z in Γ. One has

I(z) = {I}, except in the following three cases:

• z = i, in which case I(z) is the group of order 2 generated by S,

• z = ω = e

2πi/3

, in which case I(z) is the group of order 3 generated by ST ,

• z = −ω

= e

πi/3

, in which case I(z) is the group of order 3 generated by T S.

128

Theorem 6.1.3. The modular group Γ is generated by S and T .

Proof. To prove Theorems 6.1.2 and 6.1.3, consider the subgroup Γ

′

of Γ, generated by S

and T , and let z ∈ H. If G = (

a b

c d

) is an element of Γ

′

, then Im(Gz) = Im(z)/|cz + d|

Since c and d are integers, the number of pairs (c, d) such that |cz + d| is less that a

given number is ﬁn i t e. This shows tha t there exist G ∈ Γ

′

such that Im(Gz) is maximum.

Choose n ow an integer n such that T

Gz has real part between −

and

. The element

′

= T

Gz belongs to D; ind eed , it suﬃces to show that |z

′

| ≥ 1, but if |z

′

| < 1 then the

element −

′

would h ave an imaginary part strictly greater than Im(z

′

), which is im possi-

ble. Thu s, the element T

G of Γ has th e desired property. This proves (1) in Theorem 6.1.2.

We now prove assertions (2) and (3). Let z ∈ D and let G = (

a b

c d

) ∈ Γ such that Gz ∈ D.

Without lose of generality (replacing ( z, G) with (Gz, G

−1

) if necessary) we may suppose

that Im(Gz) ≥ Im(z), i. e., that |cz + d| ≤ 1. This is clearly impossible if |c| ≥ 2, leaving

the cases c = 0, 1, −1. If c = 0 , then d = ±1 and G is the tran sl at i on by ±b. Since Re(z)

and Re(Gz) are both between −

and

, this implies that either b = 0 and G = I, or

b = ±1 in which case o n e of the numbers Re(z) or Re( Gz) must be equal to −

and the

other to

. If c = 1, the fact that |z + d| ≤ 1 implies d = 0, exc ep t if z = ω (respectively if

z = −ω

), in wh i ch case we can have d = 0, 1 (respectively d = 0, −1). The case d = 0 gives

|z| ≤ 1, hence |z| = 1; on the other hand, ad − bc = 1 implies b = −1, hence Gz = a −

and the ﬁrst part of discussion proves that a = 0, except in the cases Re( z) = ±

, i. e., if

z = ω, −ω

, in which cases we have a = 0, 1 or a = 0, −1. The case z = ω, d = 1 gives

a − b = 1 and Gω = a −

ω+1

= a + ω, hence a = 0, 1. We argue similar when z = −ω

d = −1. Finally, the case c = −1 leads to the case c = 1 by changing the signs of a, b, c, d

(which does not change G seen as an element of Γ). This complet es th e veriﬁcation of (2)

and (3 ) .

To complete the proof of Theorem 6.1.3, it remains to prove that Γ

′

= Γ. Choose an

element z

∈ int D (for exam p l e z

= 2i), and let z = Gz

. We have seen above that

there exists an element G

′

∈ Γ

′

such that G

′

z ∈ D. The points z

, G

′

z = G

′

of D are

congruent modulo Γ (they li e on the same orbit in H/Γ), and one of them is interior to D.

By (2) and ( 3) of Theorem 6.1.2, it follows that these points coincide, so that G

′

G = I.

Thus, G = (G

′

)

−1

∈ Γ

′

, which completes the proof.

Corollary 6.1.4. The canonical map D → H/Γ is surjective, and its restriction to int D

is injective. In particular, D is a fundamental domain for the action of Γ on H.

Thus, D is a set intersecting each orbit of H/Γ just at one point. As a remark, one can show

that S, T | S

, (ST )

 is a presentation of Γ, or , equivalently, that Γ is the free pr oduct

of the cyclic group of order 2 genera t ed by S and th e cyclic group of order 3 generated by

ST .

129

6.2 Mod ul ar fo rm s

Deﬁnition 6.2.1. Let k be an integer. We say that a function f : H → C is weakly

modular of weight 2k if f is meromorphic on the upper half-plan e H and if it veriﬁes the

relation



az + b

cz + d



= (cz + d)

f(z), for all



a b

c d



∈ SL

(Z). (6.1)

Let G be the image in Γ of (

a b

c d

). We have

d(Gz)

= (cz + d)

−2

. The relation (6.1 ) can then

be wr it t en as

f(Gz)

f(z)



d(Gz)



−k

f(Gz)d(Gz)

= f (z)dz

, (6.2)

where dz

= dz ⊗ . . . ⊗ dz (k times). It means that the tensor product of diﬀerential 1-

forms f(z)dz

is invariant under Γ. Since Γ is g en er at ed by the elements S and T (Theorem

6.1.3), it suﬃc es to check the invariance by S and by T . This gives:

Proposition 6.2.2. Let f be a meromorphic function on H. Then, f is weakly modular

of weight 2k if, and only if, for all z ∈ H it satisﬁes the two relations:

f(z + 1) = f(z), (6.3)

f(−1/z) = z

f(z). (6.4)

Suppose the relation (6.3) is veriﬁed. We have already noticed that f can be expressed as

a function of q = e

2πiz

, function wh i ch we will denot e by

f; i t is meromorphic in the di sk

|q| < 1 wit h the o r ig in removed. We can think the valu e q = 0 making z = i∞ ∈

H, and,

f extends to a meromorphic (r espectively holomorphic) function at the origin, we say,

by abuse of language, that f is meromorphic (respectively holomorphic) at inﬁnity. This

means t h a t

f admits a Laurent expansion in a neighb o r h ood of the origin

f(q) =

∞



−∞

where th e a

’s are zero for n small en ou g h (respect i vely for n < 0).

Deﬁnition 6.2. 3. If a weakly modular function f is meromorphic at inﬁnity, we say that

f is a modular function. When f is holomorphic at inﬁnity, we set f(∞) =

f(0). This is

the val u e of f at inﬁnity.

Deﬁnition 6.2.4. A modular function which is holomorphic everywhere —including at

inﬁnity— is called a modular form; if such a function is zero at inﬁnity, it is called a cusp

form (or Spitzenform, or forme parabolique).

130

Thus, a modular form of weight 2k is given by a series f(z) =



n≥0



n≥0

2πinz

which converges for |q| < 1 (i. e., for Im(z) > 0) and that veriﬁes the identity f(−1/z) =

f(z). It is a cu sp form if a

= 0.

Example 6. 2.5 . If f and g are modular forms of weight 2k, then any C-linear combination

αf + βg is a modular form of weight 2k. Thus, the modular forms of weight 2k forms a

C-vector sp ace .

If f and f

′

are modular forms of weight 2k and 2k

′

, then the product fg is a modular form

of weight 2k + 2k

′

Example 6.2.6. We will see later that the function



n≥1

(1 − q

)

= q − 24q

+ 252q

− 1472q

+ . . . , q = e

2πiz

is a cusp form of weight 12.

6.3 Lattice functions and modular functions

Recall th at a lattice in a r ea l vector space V of ﬁnite dimension is a subgroup Λ of V

satisfying one of the following equivalent conditions:

• Λ is discrete and V/Λ is compact;

• Λ is discrete and generates the R-vector space V ;

• There exists an R-basis e

, . . . , e

of V which is a Z-basis for Λ, tha t is Λ is the n-rank

free abelian group Λ = Ze

⊕ . . . ⊕ Ze

Let L be th e set of lattices of C considered as a r e al vector space. Let M be the set of

pairs (ω

, ω

) of elements of C

such that Im





> 0. To such a pair we associate the

lattice

Λ(ω

, ω

) = Zω

⊕ Zω

with basis {ω

, ω

}. We thus obtain a map M → L which is clearly surjective. Let

G = (

a b

c d

) ∈ SL

(Z) and let (ω

, ω

) ∈ M. We put



′





a b

c d







aω

+ bω

cω

+ dω



It is clear that {ω

′

, ω

′

} is a basis for Λ(ω

, ω

). Mo r eover, if we set z =

and z

′

have

′

az + b

cz + d

= Gz.

This shows that Im(z

′

) > 0, hence that (ω

′

, ω

′

) belongs to M. In fact, we have the following

result

131

Proposition 6.3.1. For two elements of M to deﬁne the same lattice it is necessary and

suﬃcient that they are congruent modulo SL

(Z).

Proof. We just saw tha t the condition is suﬃcient. Conversel y, if (ω

, ω

) and (ω

′

, ω

′

) are

two elements of M which deﬁne the same lattice, there exists an integer matrix G = (

a b

c d

)

of determin a nt ±1 which tran sfo r m s the ﬁrst basis into the second. If det(G) < 0, the sign

of Im(z

′

) would be the opposite of Im(z

′

) as one sees by an immediate computation. The

two signs being the same, we have necessarily det(G) = 1.

Hence we can identify th e set L of lattices of C with the quotient of M/SL

(Z). Make

now C

act on L (resp ect i vely on M) by:

Λ → λΛ (respectively (ω

, ω

) → (λω

, λω

)), for λ ∈ C

The quot ie nt M/C

is identiﬁed with H by ( ω

, ω

) → z =

and th i s identiﬁcation

transforms the action of SL

(Z) on M into that of the modular group Γ = P SL

(Z) on H.

Thus:

Proposition 6.3.2. The map (ω

, ω

) →

gives by passing t o the quotient, a bijection of

L /C

onto H/Γ. (Thus, an element of H/Γ can be identiﬁed with a lattice of C deﬁned

up to a homothety.)

In numb er theory it is frequent to associate to a lattice Λ of C the elliptic curve E

= C/Λ.

It is easy to see that two lattices Λ and Λ

′

deﬁne isomorphic elliptic curves if and only

if they are homothetic. This gives a third descrip t i on of H/Γ = L /C

: it is the set of

isomorphism classes of ellipti c curves.

Let us now pass to modular functions. Let F be a function deﬁned on L , with complex

values, and let k ∈ Z. We say that F is a modular lattice function of weight 2k if

F (λΛ) = λ

−2k

F (Λ), for all lattices Λ and all λ ∈ C

. (6.5)

Let F be such a function. If (ω

, ω

) ∈ M, we denote by F (ω

, ω

) the value of F on the

lattice Λ(ω

, ω

). The formula (6.5) translates to:

F (λω

, λω

) = λ

−2k

F (ω

, ω

). ( 6 .6 )

Moreover, F (ω

, ω

) is invariant by the action of SL

(Z) on M. Formula (6.6) shows that

the product ω

F (ω

, ω

) depends only on z =

. There exists then a functio n f on H

such tha t

F (ω

, ω

) = ω

−2k

f(ω

/ω

). (6.7)

Writing that F is invariant by SL

(Z), we see that f satisﬁes the identity (6.1). Conversely,

if f veriﬁes (6.1), formula (6.7) associates to it a functio n F on L which is of weight 2k.

We can thus identify modul ar functions of weight 2k with som e lattice fun ct io n s of weight

2k.

132

Example 6.3.3 ( Ex am ple of modular functions: Eisenstein series).

Lemma 6.3.4. Let Λ be a lattice in C. The series



γ∈Λ

′

|γ|

is convergent for σ > 2. (The symbol Σ

′

signiﬁes that summation runs over all nonzero

elements of Λ).

Proof. We can proceed as with the seri es



, i. e., majorize the series unde r consideration

by a multiple of the double integral



dxdy

)

σ/2

extended over the plane without a disk

with center at 0. The double integral is co m p u t ed using the classical technique with polar

coordinates. Another equivalent method consists in remarking that the number of elements

of Λ such that |γ| is between two consecutive integers n and n + 1 is O(n); the convergence

of the series is then reduced to that of the series



σ−1

Now let k be an integer > 1. If Λ is a lattice of C, put

(Λ) =



γ∈Λ

′

. (6.8)

This series is absolut e ly convergent, thanks to Lemma 6.3.4. Observe that G

is a modular

lattice function of weight 2k. It is call ed the Eisenstein series of index k. As i n the

preceding section, we can view G

as a function on M, given by

(ω

, ω

) =



m,n

′

(mω

+ nω

)

. (6.9)

Here again t h e symbol Σ

′

means that the sum m a ti o n runs over all pairs of integers (m, n)

distinct from (0, 0). The funct io n on H corresponding to G

(by th e procedure given in

previous section) is denoted also by G

. By formulas (6.7) and (6.9), we have

(z) =



m,n

′

(mz + n)

. (6.10)

Proposition 6.3.5. Let k an integer > 1. The Eisenste in series G

(z) is a modular form

of weight 2k. We have G

(∞) = 2ζ(2k), where ζ denotes the Riemann zeta function.

Proof. The ab ove arguments show that G

(z) is weakly modular of weight 2k. We have

to show that G

is holomorphic everywhere. First suppose that z is contained in the

fundamental domain D. Then

|mz + n|

= (mz + n)(

mz + n) = (mz + n)(m¯z + n) = m

z¯z + 2mn Re(z) + n

≥ m

− mn + n

= |mω − n|

133

By Lemma 6.3.4, the series



′

|mω−n|

is conver gent. This shows that the series G

(z)

converges in D, thus also (a p p l yi ng the result to G

−1

z) with G ∈ Γ) in each of the

transforms GD of D by Γ. Since these sets cover H, we see that G

is holomorphic on H.

It remains to verify that G

is holomorphic at inﬁn i ty (and to ﬁnd the value at this po i nt).

This reduces to prove that G

has a limit for Im(z) → ∞. But one may su p pose that z

remains i n the fundamental domain D; in view of the unifor m convergence in D, we can

make the passage to t he limit term by term. The terms

(mz+n)

relative to m = 0 give 0;

the oth er s give

. Thus

lim

z→∞

(z) =



′

= 2



n≥1

= 2ζ(2k).

The Eisenstein series of lowest weights are G

and G

, which are of weight 4 and 6. It is

conveniently (because of the theory of elliptic curves) to replace these by some multiples:

= 60G

, g

= 140G

We have g

(∞) = 120ζ(4) and g

(∞) = 280ζ(6). Using the known val u es of ζ(4) and ζ(6)

(see for example Table 6.2), one ﬁnds that

(∞) =

, g

(∞) =

If we put

∆ = g

− 27g

, (6.11)

then we have ∆(∞) = 0 ; that is to say, ∆ is a cusp form of weight 12.

In fact, all these stuﬀ is related to t h e theory of elliptic curves. Let Λ be a la t ti ce of C and

let

℘

(u) =



γ∈Λ

′



(u − γ)



, (6.12)

be the correspondin g Weierstrass function. Then, G

(Λ) occur into the Laurent expansion

of ℘

℘

(u) =



k≥2

(2k − 1)G

(Λ)u

2k−2

If we put x = ℘

(u) an d y = ℘

′

(u), we have

= 4x

− g

x − g

, (6.13)

with g

= 60G

, g

= 140G

as above. Up to a numerical factor, ∆ = g

− 27g

is equal

to the discriminant of the polynomi al 4x

− g

x − g

. Usually one proves that the cubic

deﬁned by equat io n (6.13) in the projective plane is isomorphic to the ellip t i c curve C/Λ.

In part i cu l a r, it is a nonsingular curve, and th i s shows that ∆ = 0.

134

6.4 The space of modular forms

Let f be a m e ro m or p hic fu n c ti o n on H, not identically zero, and let p be a po i nt of H.

The integer n such that

(z−p)

is holomorphic and non-zero at p is called the order of f at

p and is denoted by ν

(f). When f is a modu la r function of weight 2k, th e identity (6.1)

shows that ν

(f) = ν

(f) if G is in the modular group Γ. In other terms, ν

(f) depends

only on the image of p in H/Γ. Moreover one can deﬁne ν

∞

(f) as the order for q = 0 of

the function

f(q) associ at ed to f. Finally, we will denote by e

the order of the stabilizer

of the point p; we have that e

= 2 (respectively e

= 3) if p is congruent modulo Γ to i

(respectively to ω =

−1+

√

), an d e

= 1 otherwise. Also, we have

Theorem 6.4.1. Let f be a modular function of weight 2k, not identically zero. Then

∞

(f) +



p∈H/Γ

(f) =

. (6.14)

(We can also write this formula in the form

∞

(f) +

(i) +

(ω) +



p∈H/Γ

∗

(f) =

, (6.15)

where the symbol Σ

∗

means a summation over the points of H/Γ distinct from the classes

of i and ω.)

Proof. Observe ﬁrst that the sum written in Theorem 6.4.1 makes sense, i. e., that f has

only a ﬁnite number of zeros and poles modulo Γ. Indeed, since

f is meromorphic, there

exists r > 0 such that

f has no zero nor pole for 0 < |q| < r; and this means that f has no

zero nor pole for Im(z) >

2π

log

. Now, the part D

of the fun d a m ental domain D deﬁned

by the inequality Im(z) <

2π

log

is comp a ct ; since f is meromorp hic in H, it has only a

ﬁnite number of zeros and of poles in D

, hen ce our assertion.

To prove Theorem 6.4.1, we will i ntegrate

2πi

on the bound ar y of D. More precisely:

1. Suppose that f has no zero nor pole on the boundary of D except possibly i, ω, and

−ω

. There exists a contour γ as represented in Figure 6.2, whose interior contains

a representative of each z er o or pole of f not congruent to i or ω. By the residue

theorem we have

2πi





p∈H/Γ

∗

(f).

On the other hand

• the change of variables q = e

2πiz

transforms the arc EA into a circle η centered

at q = 0, with negative orientation, and not enclosing any zero or pole of

except possibly 0. Hence

2πi



2πi



= −ν

∞

(f).

135

−1

−1/2

1/2

−¯ω

′

Figure 6.2: Integration on the boundary of region D.

• The integral of

2πi

on the circle which contains the arc BB

′

, oriented nega-

tively, has the value −ν

(f). Wh e n the radiu s of this circle tends to 0, the angle

∠B

′

tends t o

2π

. Hence

2πi



′

→ −

(f).

Similarly, when the radii of the arcs CC

′

and DD

′

tend t o 0:

2πi



′

→ −

(f), and

2πi



′

→ −

−ω

(f).

• T transforms the arc AB into the arc ED

′

; since f(T z) = f(z), we get:

2πi



2πi



′

= 0.

• S transforms the arc B

′

C onto the arc DC

′

; since f(Sz) = z

f(z), we get:

df(Sz)

f(Sz)

= 2k

df(z)

f(z)

136

hence

2πi



′

2πi



′

2πi



′



df(z)

f(z)

−

df(Sz)

f(Sz)



2πi



′

−2k

→ −2k



−1



when the radii of the arcs BB

′

, CC

′

, DD

′

tend to 0. Writing now that the

two expressi ons we get for

2πi



are equal, and passing to the limit, we ﬁnd

formula (6.15).

2. Suppose that f has a zero or a pole λ on the half line {z ∈ H : Re(z) =

, Im(z) >

√

We repeat the above proof with a contour modiﬁed in a neighborhood of λ and of T λ

as in Figure 6.3 (the arc circling a r ou nd T λ is the transfor m by T of the arc cir cl ing

around λ)

−1

−1/2

1/2

λ T λ

−¯ω

′

Figure 6.3: Integration on the boundary of region D.

We proceed in an analogous way if f has several zeros or poles o n the boundary of

D, concluding the proof.

As a remark, we on l y mention that in fact this somewhat laborious proof could have been

avoided by introducing a com plex analytic structure on the compactiﬁ c at i on

H/Γ of H/Γ,

as we mention in Chapter 1 (see for example [20]).

137

If k is an integer, we denote by M

(respectively M

) th e C-vector space of modular forms

of weight 2k (respectively of cusp forms of weight 2k). By Deﬁnition 6.2.4, M

is the kernel

of the linear form f → f(∞) on M

. Thus we h ave dim M

≤ 1. Moreover, for k ≥ 2,

the Eisenstein series G

is an elem ent of M

such that G

(∞) = 0, by Proposition 6.3. 5 .

Hence we have

= M

⊕ C G

, for k ≥ 2.

Finally, recall th at one denotes by ∆ the element g

− 27g

of M

, where g

= 60G

and

= 140G

Theorem 6.4.2. 1. We have M

= 0 for k < 0 and k = 1.

2. For k = 0, 2, 3, 4, 5, M

is a vector space of dimension 1 with basis 1, G

, G

;

we have also M

= 0.

3. Multiplication by ∆ deﬁnes an isomorphism of M

k−6

onto M

Proof. Let f be a n on ze r o element of M

. All the term s on the left side of the formula

(6.15)

∞

(f) +

(i) +

(ω) +



p∈H/Γ

∗

(f) =

are ≥ 0. Thus, we have k ≥ 0 and also k = 1, since

cannot be wri t te n in the form

n +

′

′′

, with n, n

′

, n

′′

≥ 0. This proves (1).

Now apply equation (6.15) to f = G

. We can write

in th e form n +

′

′′

, n, n

′

, n

′′

≥ 0,

only n = 0, n

′

= 0, n

′′

= 1. This shows that ν

) = 0, for p = ω (modulo Γ). The same

argument applies to G

and proves that ν

) = 1 and that all the others ν

) are zero.

This already shows that ∆ is not zero at i, h en ce is not identically zero. Since the weight

of ∆ is 12 and ν

∞

(∆) ≥ 1, formula (6.1 5 ) im p l i es th at ν

(∆) = 0 for all p = ∞ and th at

∞

(∆) = 1. In other words, ∆ does not vanish on H and has a simple zero at inﬁnity. If f

is an element of M

and if we set g = f/∆, it is clear that g is of weight 2k −12. M or eover,

the formula

(g) = ν

(f) − ν

(∆) =



(f), if p = ∞;

(f) − 1, if p = ∞;

shows that ν

(g) ≥ p for all p, thus that g belongs to M

k−6

, which proves (3). Finally,

if k < 5, we have k − 6 < 0 and M

= 0 by (1) and (3); this shows th a t dim M

≤ 1.

Since 1, G

, G

and G

are nonzero elements of M

, M

, respectively, we

have dim M

= 1 for k = 0, 2, 3, 4, 5, which proves (2).

Corollary 6.4.3. We have for any k ≥ 0

dim M



⌊k/6⌋, if k ≡ 1 (mod 6);

⌊k/6⌋ + 1, if k ≡ 1 (mod 6).

(6.16)

138

Proof. Formula (6.16) is tru e fo r 0 ≤ k < 6 by (1) and (2). Moreover, the two expressions

increase by one unit when we r ep l a ce k by k + 6 by (3). The formula is thus true for all

k ≥ 0.

Corollary 6.4.4. The space M

has for basis the family of monomials G

, with α, β ≥ 0

integers such that 2α + 3β = k.

Proof. We show ﬁrst th a t these monomials generate M

. This is clear for k ≤ 3 by (1) and

(2). For k ≥ 4, we argue by induction on k. Choose a pair (γ, δ) of integers ≥ 0 such that

2γ + 3δ = k (this is possible for all k ≥ 2). The modular form g = G

is not zero at

inﬁnity. If f ∈ M

, there exists λ ∈ C such that f −λg is a cusp form, hence equal to ∆h,

with h ∈ M

k−6

. One then applies the inductive hypothesis to h. It remains to see that

the above monomials are lin e ar l y independent; if they were not, the function G

would

verify a nontrivial algeb r ai c equati on with coeﬃcients in C, thus would be constant, which

is absu r d because G

is zero at ω but not G

As a remark, Let M =



be the graded al gebra which is the direct sum of th e M

and

let ε : C[x, y] → M be the homomorphism which maps x → G

and y → G

. Corollary

6.4.4 is equivalent to saying that ε is an isomorphism. Hence, one can identify M with the

polynomial algebra C[G

, G

6.5 The mod ul ar inva ri a nt j

We deﬁne the j function as

j(z) = 1728

(z)

∆(z)

1728g

(z)

(z) − 27g

(z)

. (6.17)

Proposition 6.5.1. The function j is a modular function of weight 0. Moreover, it is

holomorphic on H and has a simple pole at inﬁnity. It deﬁnes by passage to quotient a

bijection of H/Γ onto C.

Proof. First assertion comes from the fact that g

and ∆ ar e bo t h of weight 12. The

holomorphicity of j follows from the fact that ∆ = 0 on H and has a simple pole at inﬁnity,

while g

is nonzero at inﬁnity. To pr ove the last assertion, on e has to show that if λ ∈ C,

the m odular form f

= 172 8g

− λ∆ has a unique zero modulo Γ. To see this, one applies

formula (6.15) to f

and k = 6 . The only decompositions of

= 1 in the form n +

′

′′

with n, n

′

, n

′′

≥ 0 correspond s to

(n, n

′

, n

′′

) = (1, 0, 0) or (0, 2, 0) or (0, 0, 3) .

This sh ows that f is zer o at one and only one point of H/Γ.

Theorem 6.5.2. Let f be a meromorphic function on H. The following are equivalent:

139

1. f is a modular function of weight 0;

2. f is a quotient of two modular forms of the same weight;

3. f is a rational function of j.

Proof. The implications (3) ⇒ (2) ⇒ (1) are immediat e. We show that (1) ⇒ (3). Let f be

a modular funct i on. Being free to multiply by a su i t ab l e polynomial in j, we ca n suppose

that f is holomorph ic on H. Since ∆ is zero at inﬁnity, there exists an integer n ≥ 0

such that g = ∆

f is h ol om o r p h i c at inﬁn i ty. The function g is then a modular form of

weight 12n. By Corollary 6.4.4, we can writ e g as a linear combination of the G

, with

2α + 3β = 6n. By linear i ty, we are red uced to the case g = G

, i. e., f = G

/∆

But the rela t io n 2α + 3β = 6n shows that p =

and q =

are integers and one has

f = G

/∆

p+q

. Thus, we only need to see that G

/∆ and G

/∆ are rational functions

of j, which is obvious from the deﬁnitions of g

, g

and ∆.

As we stated in Chapter 1, it is possible to deﬁne in a natural way a structure of com p l ex

analytic manifold on th e compactiﬁcation

H/Γ of H/Γ. Proposition 6.5.1 thus means that

j deﬁnes an isomorph is m of

H/Γ onto the Riemann sphere C = C ∪{∞}. As for Theorem

6.5.2, it amou nts to the well known fact that the only meromorphic functions on

C are the

rational functions.

The coeﬃcient 1728 = 2

· 3

has been introduced in or d e r that j has a resi d u e equal to 1

at inﬁnity. More precisely, the series expansion of n ex t section shows t h a t the q-expansion

of j (recall equation (1.3)) is

j(z) = q

−1

+ 744 +



n≥1

, (6.18)

where q = e

2πiz

, z ∈ H. One has c

= 2

·3

·1823 = 196884, c

= 2

·5·2099 = 21493760.

All c

are integers (this follows from the deﬁnition of j and the q-expansion formulas of

, g

), an d they enjoy remarkable divisibility properties, see for example [2 ] or [38].

6.6 Expansions at inﬁnity

Eisenstein series and the j function are closely related to the Riemann zeta function. Only

to give a basic idea, we will present some of this results, omitting most of the proofs.

Interested reader can consult [117], [167].

Consider the Bernoulli numbers B

, deﬁn ed by the series

− 1

= 1 −



k≥1

(−1)

k+1

(2k)!

. (6.19)

140

691

2730

3617

510

43867

798

283·617

330

Table 6.1: First Bernoulli numbers B

There are a lot of properties rel at ed to Bernoulli numbers, (see [101 ] ) . For example, the

give the values of the Riemann zeta function for th e positive even integers ( an d also for

the neg at i ve odd integers):

Proposition 6.6.1. If k ≥ 1 is an integer, then

ζ(2k) =

2k−1

(2k)!

. (6.20)

Proof. The identity

z cot z = 1 −



k≥1

(2k)!

(6.21)

follows from the deﬁnition of the B

by putting x = 2iz. Moreover, ta ki n g the logarithmic

derivative o f

sin z = z



n≥1



1 −



, (6.22)

we get:

z cot z = 1 + 2



n≥1

− n

= 1 − 2



n≥1



k≥1

. (6.23)

Comparing (6.21) and (6.23 ) , we get (6.20).

Table 6.2 shows some values of the Riemann ζ function

ζ(2) =

2·3

ζ(4) =

2·3

·5

ζ(6) =

·5·7

ζ(8) =

2·3

·5

·7

ζ(10) =

·5·7·11

ζ(12) =

691π

·5

·7

·11·13

ζ(14) =

2π

·5

·7·11·13

. . .

Table 6.2: Some values ζ(2k) for the Riemann ζ function.

We now give the Taylor expansion of t h e Eisenstein series G

(z) with respect to q = e

2πiz

Let us start with the well known formula

π cot πz =



m≥1



z + m

z − m



We have on the other hand

π cot πz = π

cos πz

sin πz

= πi

q + 1

q − 1

= πi −

2πi

1 − q

= πi − 2πi



n≥0

141

and com p ar i n g , we get:



m≥1



z + m

z − m



= πi − 2πi



n≥0

. (6.24)

By successi ve diﬀerentiations of (6.24), we obtain t h e following formula (valid for k ≥ 2 )



m∈Z

(z + m)

(k − 1)!

(−2πi)



n≥1

k−1

. (6.25)

Denote now by σ

(n) =



d|n

the sum of k-th powers of positive divisors of n.

Proposition 6.6.2. For every integer k ≥ 2, one has:

(z) = 2ζ(2k) + 2

(2πi)

(2k − 1)!



n≥1

2k−1

(n)q

. (6.26)

Proof. To prove this, we expand G

(z) =



(n,m)

′

(nz+m)

= 2ζ(2k) + 2



n≥1



m∈Z

(nz+m)

Applying (6.25) with z replaced by nz, we get

(z) = 2ζ(2k) + 2

2(−2πi)

(2k − 1)!



d≥1



a≥1

2k−1

= 2ζ(2k) + 2

(2πi)

(2k − 1)!



n≥1

2k−1

(n)q

Deﬁnition 6.6.3. For k ≥ 1, we deﬁne the Eisenstein series E

(z) by

(z) = G

(z)/2ζ(2k), (6.27)

where G

(z) is the Eisenstein series of index k deﬁned in (6.8) .

Corollary 6.6.4. For k ≥ 2, one has

(z) = 1 + γ



n≥1

2k−1

(n)q

, (6.28)

where γ

= (−1)

Proof. When taki n g the quotient of G

(z) by 2ζ(2k) in equation (6.26), it is clear that

(z) is given by (6.28). Th e coeﬃcient γ

is compu t e d using Proposition 6.6.1

(2πi)

(2k − 1)!

2ζ(2k)

(2π)

(−1)

(2k − 1)!

(2k)!

2k−1

= (−1)

142

= 1 + 240



n≥1

(n)q

= 1 − 504



n≥1

(n)q

= 1 + 480



n≥1

(n)q

= 1 − 264



n≥1

(n)q

= 1 +

65520

691



n≥1

(n)q

= 1 − 24



n≥1

(n)q

Table 6.3: Expan si o n of the ﬁrst Eisenstein series E

Table 6.3 gives the q-expansion of some Eisenstein series E

Since g

= 60G

, g

= 140G

, we have

60(2π

)

2 · 3

· 5

(2π)

· 3

, (6.29)

140(2π

)

· 5 · 7

(2π)

· 3

. ( 6. 30)

Recall now that

∆ = g

− 27g

(2π)

· 3

− E

) = (2π)



q − 24q

+ 252q

− 1472q

− . . .



In fact, we can write a co m p ac t form for the last expansion , due to Jacobi. We will not

prove thi s here. The reader can look [117] or [167].

Theorem 6.6.5 (Jaco b i ) .

∆ = (2π)



n≥1

(1 − q

)

. (6.31)

Note that the cusp form η(z) = (2π)

−12

∆(z) = q



n≥1

(1 − q

)

, is the Dedekind fun ct i on

mentioned in Example 1 .4 .2. Usually τ(n) denote the coeﬃcient of q

in the expansion of

η(z), thus

η(z) = q



n≥1

(1 − q

)



n≥1

τ( n) q

The function n → τ (n) is cal l ed the Ramanujan’s function. Observe also that from the

deﬁnition of j (6.17) and equations (6.29), (6.31) we can rewrite

j = 1728

∆

= 1728



(2π)



(2π)

= 1728

(2π)

1728

(2π)

143

so we obtain the expression (1.3) for j given in Chapter 1.

A useful estimate for the coeﬃ ci ents of modular functions, which s h ows that the quotient

|/n

remains bounded when n → ∞, is given in the following result due to Hecke. For

a proof see [165, p.94]

Theorem 6.6.6. If f =



is a cusp form of weight 2k, then a

= O(n

Remark 6.6.7. The theo r y of quasi-modular forms extends the cl ass ica l theory of modular

forms, when it is equipped with the diﬀerencial ∂/∂z. See for example [152].

6.7 Theta functions

Let V be a real vector space of ﬁnite dimension n endowed with an invariant measure µ.

Let V

∗

be the dual of V . Let f be a rapidly decreasing smooth function on V (see[163]).

The Four ier transform

f of f is deﬁned by the formula

f(y) =



−2πix,y

f(x) dµ(x). (6.32)

This is a rapidly decreasing smooth function on V

∗

. Let now Λ be a lattice in V

∗

. We

denote by Λ

′

the lattice in V

∗

dual to Λ; that is the set of y ∈ V

∗

such that x, y ∈ Z, for

all x ∈ Λ. Observe that Λ

′

may be identiﬁed with t h e Z-dual of Λ (hence the terminology).

Proposition 6.7.1 (Poisson formula). Let ν = µ(V/Λ). One has:



x∈Λ

f(x) =



y∈Λ

′

f(y).

After replacing µ by ν

−1

µ, we can assume that µ(V/Λ) = 1. By taking a basis e

, . . . , e

Λ, we identify V with R

, Λ with Z

, and µ with the p r oduct measure dx

···dx

. Thus

we have V

∗

= R

, Λ

′

= Z

and we are reduced to the classical Poisson formula (see [163])

We suppose now that V is endowed with a symmetric bilinear form x, y which is positi ve

and non degenerate (i. e., x, x > 0 if x = 0). We identify V with V

∗

by means of this

bilinear form. The lattice Λ becomes thus a lattice in V ; and one has y ∈ Λ if and only if

x, y ∈ Z, for all x ∈ Λ. To a lattice Λ, we associate the following functio n deﬁned on R

(t) =



x∈V

−πtx,x

. (6.33)

We choose th e invari ant measure µ on V such that, if e

, . . . , e

is an or t h on o r m al basis

of V , the unit cube deﬁned by the e

has volume 1. The volume ν of th e lattice Λ is th e n

deﬁned by ν = µ( V /Λ).

144

Proposition 6.7.2. We have the identity

(t) = t

−n/2

−1

′

−1

Proof. Let f = e

−πx,x

. It is a rapidly d ecr ea sing smooth function on V . The Fourier

transform

f of f is equal to f. Ind ee d , choose an orthon o r m al basis of V and use this

basis to identify V with R

; the measure µ becomes the measure dx = dx

···dx

and the

function f is

f = e

−π(x

+...+x

)

We are thus reduced to showing that the Fourier tr an sfo rm of e

−πx

is e

−πx

, which is well

known. We now apply Proposit i on 6.7.1 to the function f and to the lattice t

1/2

Λ; th e

volume of this lattice is t

n/2

v and its dual is t

−1/2

′

; hen ce we get the formula.

We can give a matrix interpretation. L et e

, . . . , e

be a basis of Λ. Put a

= e

, e

. The

matrix A = [a

] is positive, non degenerate and symmetric. If x =



is an element of

V , then

x, x =



i,j

The fun ct i on Θ

can be writt en as

(t) =



∈Z

−πt



. (6.34)

The volume ν of Λ is given by ν = (det A)

1/2

. This can be seen as follows: Let ε

, . . . , ε

be an orthon or m al basis of V and put

ε = ε

∧ . . . ∧ ε

, e = e

∧ . . . ∧ e

We have e = λε, with |λ| = ν. Moreover, e, e = det Aε, ε, and by comparing, we obtain

= det A. Let B = [b

] be the matrix inverse to A. One checks immediately that the

dual b a si s {e

∗

} to {e

} is given by the formulas

∗



, for i = 1, . . . , n.

The {e

∗

} form a basis of Λ

′

. The matrix [e

∗

, e

∗

] is equal to B. Th i s shows in particular

that i f ν

′

= µ(V /Λ

′

), then we have νν

′

= 1.

We will be interested in pairs (V, Λ) wh ich have the following two properties:

1. The dual Λ

′

of Λ is equal to Λ.

2. We have x, x ≡ 0 (mod 2), for all x ∈ Λ.

145

Condition (1) says that one has x, y ∈ Z, for all x, y ∈ Λ and that the form x, y

deﬁnes an isomorphism of Λ onto its dual. In m a t r ix terms, it means that the mat r i x

A = [e

, e

] has integer coeﬃcients and that its determinant equals 1, or equivalently, to

ν = 1. Condition (2) means that the diagonal terms of A are even. We have cal le d these

lattices even unimodular (see Section 4.4). Suppos e that the pair (V, Λ) satisﬁes condi t i on s

(1) and (2) above, that i s, Λ is even unimodular. Let m ≥ 0 be an integer, and denote by

the set of elements x ∈ Λ such that x, x = 2m. It can be seen that |Λ

| is bounded

by a polyn om i al in m (a crude volume argument gives for instance |Λ

| = O(m

n/2

)). This

shows that the series with integer coeﬃcients

(q) =



x∈Λ

x,x/2



m∈Z

, (6.35)

deﬁned in equation (4.29), converges for |q| < 1. Thus one can deﬁne a function θ

(z) on

the half-plane H by the formula ( 6. 35), with q = e

2πiz

. Th e function θ

is called the theta

function of the quadratic m odule Λ. It is holomorphic on H.

Theorem 6.7.3. 1. The dim ension n of V is divisible by 8.

2. Also, the function θ

is a modular form of weight n/2.

Proof. We prove th e identity

(−1/z) = (iz)

n/2

(z). (6.36)

Since the two sides are analytic in z, it suﬃces to prove this formula when z = it with

t > 0 real. We have

(it) =



x∈Λ

−πtx,x

= Θ

(t).

Similarly, θ

(−1/it) = Θ

−1

). Formula (6.36) res u l t s thus from Proposition 6 .7 . 2 , takin g

into accou nt that ν = 1 and Λ = Λ

′

Now, to prove th e ﬁrst assertion, su p pose that n is not divisibl e by 8; replacing Λ, if

necessary, by Λ ⊕ Λ or Λ ⊕ Λ ⊕ Λ, we may suppose that n ≡ 4 (mod 8). Formula (6.36)

can th en be written

(−1/z) = (−1)

n/4

m/2

(z) = −z

n/2

(z).

If we put ω(z) = θ

(z) dz

n/4

, we see that the diﬀerential form ω is transformed into −ω by

S : z → −1/z. Since ω is invariant by T : z → z + 1, we see that ST tra n sfor m s ω into

−ω, which is absu r d because (ST )

= 1.

For (2), since n is divisible by 8, we can rewrite ( 6 .3 6 ) in the form

(−1/z) = z

n/2

(z) (6.37)

which shows that θ

is a modular form of weight

146

Corollary 6.7.4. There exists a cusp form f

of weight

such that θ

= E

+ f

, where

k =

Proof. This follows from th e fact that θ

(∞) = 1, hence that θ

− E

is a cusp form.

Corollary 6.7.5. We have |Λ

| =

2k−1

(m) + O(m

), where k =

Proof. This follows from Corol la r y 6.7.4, equation (6.28) and Theorem 6.6.6.

Note t h at the ‘error term’ f

is in general nonzero. However, Siegel has proved th at the

weighted mean of the f

is zero. More preci sel y, let C

be the set of classes (up to isomor-

phism) of even unimodular lattices Λ, and denote by g

the order of the automorphism

group of Λ ∈ C

. One has:



Λ∈C

· f

= 0, (6.38)

or equi valently



Λ∈C

· θ

= M

, wh er e M



Λ∈C

. (6.39)

Note that this is also equivalent to saying that the weighted me an o f the θ

is an eigen-

function of the Hecke operator T (n) (for a proof see [ 16 5 ] ).

Example 6.7.6 (The case n = 8). Every cusp form of weight

= 4 is zero. Corollary

6.7.4 then shows that θ

= E

, in other words, |Λ

| = 240σ

(m) for all integers m ≥ 1.

This ap p l i es to the lattice Q

constructed in S ect i on 4.6.

Example 6.7.7 (The case n = 16). For the same reason as above, we have θ

= E

1 + 480



(m)q

. Here one may take Λ = Q

⊕ Q

or Λ = Q

; even though these

two lattices are not isomorphic, they have the same theta function , i. e., they represent

each integer the same number of t i m es. Note that the function θ associated to the lattice

⊕ Q

is the square of the function θ of Q

; we recover thus the identity:



1 + 240



m≥1

(m)q



= 1 + 480



m≥1

(m)q

Example 6.7.8 (The case n = 24). The space of modular forms of weight 12 is of dimension

2. It has for basis the two functions:

= 1 +

65520

691

(m)q

and

η = (2π)

−12

∆ = q



m≥1

(1 − q

)



m≥1

τ( m)q

147

The theta fun c ti o n associated wit h the lattice Λ can t hus be written θ

= E

+ c

η, with

∈ Q. We have

|Λ

| =

65520

691

(m) + c

τ( m), for m ≥ 1.

The coeﬃcient c

is deter m i n ed by putting m = 1:

= |Λ

| −

65520

691

, for m ≥ 1.

For exampl e:

• The Leech lattice Λ

is such th a t |(Λ

)

| = 0 (Λ

has no 1-norm elements). Hence,

= −

65520

691

• The lattice Q

⊕ Q

is such that |(Q

)

| = 3240. Hence, c

432000

691

148

Chapter 7

Conway-Norton fundamental

conjecture

This chapter is a preamble to the proof given by Borcherds. In fact, we d evelop some

techniques and tools which appear in the Conway-Norton conjecture. We i n i t i al ly give

some ba ckground theory on Hecke operators. Then we descr i be som e properties of the

congruence su b gr oups of SL

(Z) and their normalizers. Fi n al l y, following [38], we mention

various moonshine properties that leaded Conway and Norton to formulate their conjectu r e,

including the r ep l i cat i on formulas, useful in the proof of Moo n sh i n e conjecture.

7.1 Hecke operators

Deﬁnition 7.1.1. Let E be a set and let X

be the free abelian grou p generated by E.

A correspondence on E (with integer coeﬃcients) is a homomorphism T : X

→ X

. We

can give T by its values o n the elements x of E:

T (x) =



y∈E

(x)y, where n

(x) ∈ Z, (7.1)

the n

(x) being ze r o for almost all y.

Let F be a numerical valued function on E. By Z-linearity, it extends to a function (again

denoted F ), on X

. The transform of F by T , denoted T F, is the restriction to E of th e

function F ◦ T . With the notations of (7.1), we have

T F(x) = F (T(x)) =



y∈E

= n

(x)F (y). ( 7. 2)

Example 7.1.2 (The operators T (n)). Let L be the set of lattices of C (c. f. Section 6.3) .

Let n ≥ 1 be an integer. We denote by T (n) the correspondence on L which transforms

a latti ce to the sum (in X

) of its sublattices of index n. Thus we have:

T (n)Λ =



[Λ:Λ

′

]

′

, i f Λ ∈ L . (7.3)

The sum on the right-hand side of (7.3) is ﬁnite. Indeed, the latti ces Λ

′

all contain nΛ and

their number is also the number of subgroups of order n of Λ/nΛ = (Z/nZ)

. If n is prime,

149

one sees in fact that thi s number is equal to n + 1 (number of points of the projective lin e

over a ﬁel d with n elements). We also use the homothety operato r s R

(λ ∈ C

), deﬁned

Λ = λΛ, if Λ ∈ L . ( 7. 4)

Deﬁnition 7.1.3. The operators T (n) deﬁned above are usually called Hecke operators.

It makes sense to compose the cor r espondences T (n) an d R

, since they are endomorphisms

of the abelian group X.

Proposition 7.1.4. The correspondences T (n) and R

verify the identities

1. R

= R

λµ

, for all λ, µ ∈ C

2. R

T (n) = T (n)R

, for all n ≥ 1 and all λ ∈ C

3. T (m)T (n) = T (mn), if (m, n) = 1.

4. T (p

)T (p) = T (p

n+1

) + pT (p

n−1

, for p prime, n ≥ 1.

Proof. Statements (1) and (2) are trivial. Note that (3) is equivalent to the following

assertion: Let m, n ≥ 1 be two relati vely prime integers, and let Λ

′′

be a su b l at t i ce of a

lattice Λ of index mn; there exists a unique sublatt i ce Λ

′

of Λ, containi n g Λ

′′

, su ch that

[Λ : Λ

′

] = n and [Λ

′

: Λ

′′

] = m. This assertion follows itself from the fact that the group

Λ/Λ

′′

, which is of order mn, decomposes uniquely into a direct sum of a group of order

m and a group of order n (Bezout’s theorem). To prove (4), let r be a lattice. Then

T (p

)T (p)Λ, T (p

n+1

)rΛ and T (p

n−1

Λ are linear combinatio n s of latt i ces contained in

Λ and of index p

n+1

in Λ (note that R

Λ is of index p

in Λ ) . L et Λ

′′

be such a lattice. In

the above lin ea r combinations it appear s with coeﬃcients a, b, c, say; we have to show that

a = b + pc, i. e., that a = 1 + pc since b is equ al to 1.

We have two cases: (i) Λ

′′

is not contained in pΛ. Then c = 0 and a is the numbe r of

lattices Λ

′

, intermediate between Λ and Λ

′′

, and of index p i n Λ; such a lattice Λ

′

contains

pΛ. In Λ/pΛ the image of Λ

′

is of index p and it contains the image of Λ

′′

, which is of order

p (hence also of index p because Λ/pΛ is of order p

); hence there is only one Λ

′

which does

the trick. This gives a = 1 and the formula a = 1 + pc is valid. (ii) Λ

′′

⊆ pΛ. We have

c = 1; any lattice Λ

′

of index p in Λ contains pΛ, thus a fortiori Λ

′′

. This gives a = p + 1

and a = 1 + pc is again valid.

Corollary 7.1.5. For n > 1, the T (pn) are polynomials in T (p) and R

This foll ows from (4) by induction on n. Moreover,

Corollary 7.1.6. The algebra generated by the R

and the T (p), p prime, is commutative;

it contains all the T (n).

150

We will study now the act i on of T (n) on the lattice functions of weight 2k. Let F be a

function on L of weight 2k (recall Section 6.3). By deﬁnition

F = λ

−2k

F, for al l λ ∈ C

. (7.5)

Let n be an integer. Property (2) in Propositi on 7.1.4 shows that

(T (n)F) = T (n)(R

F ) = λ

−2k

T (n)F,

in other words, T (n)F is also of weight 2k. Assertions (3) and (4) on same proposition

give:

T (m)T(n)F = T (mn)F, if (m, n) = 1; (7.6)

T (p)T(p

)F = T (p

n+1

)F + p

1−2k

T (p

n−1

)F, for p prime, n ≥ 1. (7.7)

Let Λ be a lattice with basis {ω

, ω

} and let n ≥ 1 be an integer. The following lemma

gives all the sublattices of Λ of ind ex n:

Lemma 7.1.7. Let S

be the set of integer matrices (

a b

0 d

), with ad = n, a ≥ 1, 0 ≤ b < d.

If σ = (

a b

0 d

) is contained in S

, let Λ

be the sublattice of Λ having for basis

′

= aω

+ bω

, ω

′

= dω

The map σ → Λ

is a bijection of S

onto the set Λ(n) of sublattices of index n in Λ.

Proof. The fact th at Λ

belongs to Λ(n) foll ows from th e fact t h a t det σ = n. Conversely

let Λ

′

∈ Λ(n) . We put

= Λ/(Λ

′

+ Zω

) and Y

= Zω

/(Λ

′

∩ Zω

These are cyclic groups generated respectively by the i m ag es of ω

and ω

. Let a a n d d be

their o r d er s. The exact sequence

0 −→ Y

−→ Λ/Λ

′

−→ Y

−→ 0

shows that ad = n. If ω

′

= dω

, then ω

′

∈ Λ

′

. On the other hand, there exists ω

∈ Λ

′

such tha t

′

≡ aω

(mod Zω

It is clear that ω

′

and ω

′

form a basis of Λ

′

. Moreover , we can write ω

′

in th e form

′

= aω

+ bω

, wit h b ∈ Z,

where b i s uniquely determined modul o d. If we impose on b the inequ a li ty 0 ≤ b < d, this

ﬁxes b, and also ω

′

. Thus, we have associated to every Λ

′

∈ Λ(n) a matrix σ(Λ

′

) ∈ S

, an d

one checks that the maps σ → Λ

and Λ

′

→ σ(Λ

′

) are inverses to each other; the lemma

follows.

Example 7.1.8. If p is a prime, the elements of S

are the matrix



p 0

0 1



, and the matrices



1 b

0 p



, with 0 ≤ b < p.

151

7.2 Action of T (n) on modular functi o ns

Let k b e an integer, and let f be a weakly modular function of weight 2k. As we saw in

Section 6.3, f correspon d s to a function F of weight 2k on L , satisfying equation (6.7)

F (Λ(ω

, ω

)) = ω

−2k

f(ω

/ω

). (7 .8 )

We deﬁne T (n)f as the function on H associated to the funct i on n

2k−1

T (n)F on L . (Note

that the numerical coeﬃcient n

2k−1

gives formulas ‘without denom i n at or s ’ in what follows.)

Thus by deﬁnition:

T (n)f(z) = n

2k−1

T (n)F(Λ(z, 1)),

or else by Lemma 7.1.7

T (n)f(z) = n

2k−1



ad=n, 0≤b<d



az + b



. (7.9)

Proposition 7.2.1. The function T (n)f is weakly modular of weight 2k. It is holomorphic

on H if f is. We have:

T (m)T(n)f = T (mn), if (m, n) = 1, (7.10)

T (p)T(p

)f = T (p

n+1

)f + p

2k−1

T (p

n−1

)f, if p is prime, n ≥ 1. (7.11)

Proof. Formula (7.9) shows that T (n)f i s meromorphic on H, thus weakly m odular; if in

addition f is holomorphic, so is T (n)f. Formulas ( 7. 10) and (7. 11) foll ow from formulas

(7.6) and (7.7) taking int o account the numerical coeﬃcient n

2k−1

incorporated into t h e

deﬁnition of T (n)f.

Now suppose tha t f is a modular function (of weight 0), with q-ex p an s io n of the form

f(z) =



m∈Z

, q = e

2πiz

Theorem 7.2.2. T (n)f is also a modular function with q-expansion

T (n)f(z) =



m∈Z



s|(m,n)

−1

mn/s

Proof. By deﬁnition, we have

T (n)f(z) = n

−1



ad=n, 0≤b<d



µ∈Z

2πiµ(az+b)/d

The sum



0≤b<d

2πiµb/d



0, µ ≡ 0 (mod d);

d, µ ≡ 0 (mod d);

152

since it is the sum 1

+ζ

+. . .+ζ

over all d-roots of un i ty. Hence the sum is over multiples

of d. Puttin g µ = td,

T (n)f(z) = n

−1



ad=n, µ=td

2πiaµz/d

= n

−1



ad=n, t∈Z

Collecting powers of t and putting m = at, this gives

T (n)f(z) =



m∈Z



a|(m,n)







m∈Z



a|(m,n)

−1

mn/a

Since f s m er om or phic at inﬁnity, there exists an integer N ≥ 0 such that c

= 0 for all

m ≤ −N. The c

md/a

are thus zero for all m ≤ −nN, which shows that T (n)f is also

meromorphic at inﬁnity. Since it is weakly modular (by Proposition 7.2.1), it is also a

modular function.

7.3 Congruence subgroups of SL

(R)

We shal l describe in this section the genus 0 subgroups G ⊆ SL

(R) which appear in t he

Conway-Norton co n ject u r e (see Chapter 1).

Let N be a positive integer an d let

(N) =



a b

c d



∈ Γ : c ≡ 0 (mod N )



This is called a congruence subgroup of SL

(Z). We shall describe the normalizer

(R)

(Γ

(N)),

i. e., the largest subgroup of SL

(R) in which Γ

(N) is normal. In describing this norm al -

izer, the divisors of 24 play a crucial role. Let h be the largest divisor of 24 such t h a t h

divides N, and consider the factorization

N = hn.

Here, n is a positive integer d i v is ible by h. Let T be the subg ro u p of SL

(R) given by

T =



a b/h

cn d



∈ SL

(R) : a, b, c, d ∈ Z, ad − bcn/h = 1



It is readily seen that T i s a subgroup of SL

(R) containing Γ

(N) and that Γ

(N) is

normal in T . Hence

T ⊆ N

(R)

(Γ

(N)).

153

In fact, T is conjugate to Γ

(n/h) in SL

(R) sinc e we have



1/2

0 h

−1/2



a b/h

cn d



−1/2

0 h

1/2





a b

cn/h d



Now, T is not in general th e full normalizer of Γ

(N). However, T is normal in this

normalizer and the factor group is an e lem e ntary abelian 2-group. This can be understood

as follows.

Deﬁnition 7.3.1. A Hall divisor of n/h is a divisor e such that ( e, n/he) = 1.

The number of Hall divisors of n/h is a power of 2. The Hall divisors form an elementary

abelian group Z

× . . . × Z

under the composition e ⋆ f = g, where

g =

(e, f)

For each Hall divisor e of n/h we may describe a coset of T in the norma l ize r of Γ

(N),

which corresponds to e under the above isomor phism. This coset consists of all mat r i ces of

the form



1/2

(b/h)e

−1/2

cne

−1/2

1/2



for a, b, c, d ∈ Z and ade −bc(n/h)e

−1

= 1. Moreover, N

(R)

(Γ

(N)) is the union of these

cosets for all Hall d i v is or s e of n/h (see [38]). Thus, the quotient of the norm a li zer by its

normal subgroup T is isomor p h i c to the group of Ha l l divisors of n/h. Thus we have

(R)

(Γ

(N))/T

∼

× . . . × Z

and th e number of factors on the right-hand side is the number of distinct prime divisors

of n/h. We can now st a t e a more precise form of the Conway-Norton conjecture 1.7.1

Conjecture 7.3.2 (Moonshine Conjecture). (Conway-Norton, 1979). Let g be an ele-

ment of the Monster group M. Then, the McKay-Thompson series T

(z) is the normalized

Hauptmodul

(z) :

H/G → CP

for some genus 0 subgroup G of SL

(R) satisfying

T ⊆ G ⊆ N

(R)

(Γ

(N)),

for some N.

We call a discrete subgroup G of SL

(R) a sub gr ou p of moonshine-type if it contains some

congruence subgroup Γ

(N), and obeys



1 t

0 1



∈ G ⇒ t ∈ Z. (7.12)

154

Of course, not all subgroups G satisfying the above condition will have genus 0. Thompson

[173] proved that there are only ﬁnitely many modular grou p s of moonshine-type in each

genus. Cumm ins [42] has found all of th es e of genus 0 and 1. In particular , t h er e are

precisely 6,486 genus 0 m oonshine-type group s. Exactly 616 of these h ave Hauptmoduls

with rational (in fact integral) coeﬃci ents; the remainder have cyclotomic integer coeﬃ-

cients. There are some natural equivalences (for exampl e a Galois action) which collapse

this number to 371, 310 of which have integral Hauptmoduls.

We i ll u s t ra t e this situation by considering the special case in which N is prime. So, let

N = p be a prime, and then we have n = p and h = 1. It follows that T = Γ

(p) an d

(R)

(Γ

(p)) : T ] = 2.

Now, it has been shown by Ogg (see [158]) that N

(R)

(Γ

(p)) h as genus 0 if an d only if

p ∈ {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 },

the pr i m e divisors of the order of the Monster group M (recall Chapter 1).

As mentioned in the introductory chapter, th e central stru ct u r e in th e attem p t t o un der-

stand equations (1. 7 ) is an in ﬁ n i t e- d i m en si o n al Z-graded module for the Monster, V

♮

−1

⊕V

⊕. . ., with graded dimension J(z) (see (1.8)). If we let ρ

denote the n-th small-

est irreducible M-module, with dimension d

numbered as in Table 1.2, then t h e ﬁrst few

subspaces will be V

= ρ

, V

= {0}, V

= ρ

⊕ρ

, V

= ρ

⊕ρ

, V

= 2ρ

⊕2ρ

⊕ρ

and so on. As we know from the rep r esentation theor y for ﬁnite groups, a dime n si on can

(and should) be replaced with the character. This gives us the graded traces (1.9)

(z) =



n∈Z

tr(g|V

or McKay-Thompson series for this module V . Of course, T

= J.

We write c

(n) to be the coeﬃcient of q

in McKay-Thompson series T

, tha t is

(z) = q

−1



n≥1

(n)q

, wit h q = e

2πiz

Conway-Norton conjecture (7.3.2) states th at for each element g of the Monster M, the

McKay-Thompson seri es T

is the Hauptmodul J

(z) for a genus 0 g ro u p G

⊆ Γ of

moonshine-type (recall ( 7. 12)). These groups each contain Γ

(N) as a normal subgroup ,

for some N dividing o(g) ·(24, o(g)), o(g) th e order of g, and the quotient grou p G

/Γ

(N)

have exponent ≤ 2. So for each n the map g → c

(n) is a character tr(g|V

) of M. Conway

and Nor t o n [38] explicitly identify each of the groups G

. The ﬁr st 50 coeﬃcients c

(n) of

each T

are given in [148]. Together with the recursions given in Section 7.5 below, this

allows one to eﬀectively com pute arbitrarily many coeﬃcients c

(n) of the Hauptmoduls.

155

It is also this that uniquely deﬁnes V

♮

, up to equivalence, as a graded M-module.

There are around 8×10

elements in the Monster, so naively we may ex pect about 8×10

diﬀerent Hauptmoduls T

. However, a character evaluated at g and at any of his conjugat es

hgh

−1

will always be equal, so T

= T

hgh

−1

. Hence there can be at most 19 4 distinct T

(one for each conjugacy class). All coeﬃcients c

(n) are integers ( a s are in fact most entries

of the charact er table of M). This implies that T

= T

whenever the cyclic subgroups

g and h are equal. In fact, the total numbe r of distinct McKay-Thompson series T

arising in Monstrous Moonshine turns out to b e only 171. Of those many redundan ci es

among the T

, only one is unexpect ed —and unexplained—: the McKay-Thompson series

of two unrelated classes of o rder 27, nam e ly 27A and 27B (i n Atlas notation), are equal.

It would be interesting to understand what gen er al phenomenon, if any, is res ponsible for

27A

= T

27B

. But as we know from the theory of vertex algebras, the McKay-Thompson

series T

(z) are actu al l y specialisations of 1-point fu n ct i ons and as such are fun ct i on s of not

only z but of all M- i nvariant vectors v ∈ V

♮

. What we call T

(z) is really the spe cia l isa t io n

(z, 1) of this function T

(z, v).

Not all subgroups G of genus 0 satisfyi n g T ⊆ G ⊆ N

(R)

(Γ

(N)), for some N, correspon d

to McKay-Thompson series. If h = 1 almost all of them do, but there a r e three exceptions

(c. f. [38]). It has recently shown by Conway (see [105]) that, apar t from these exceptions

when h = 1, there is a fairly simpl e characterization of the groups ar i si n g as G

in Monstrous

Moonshine:

Proposition 7.3.3. A subgroup G of SL

(R) equals one of the modular groups G

appear-

ing in Conjecture 7.3.2 if, and only if

1. G is genus 0;

2. G has the form Γ

(n/h) + e, f, g, . . .;

3. the quoti ent of G by Γ

(nh) is a group of exponent ≤ 2;

4. each cusp Q ∪{i∞} can be mapped to {i∞} by an element of SL

(R) that conjugates

the group to one containing Γ

(nh).

This is an observation mad e by examining the possible cases, although the signiﬁc an ce of

this condition is not yet underst ood. Also, n o t at i on Γ

(n/h) + e, f, g, . . . in (2) corresponds

to the full normaliser Γ

(N) in P SL

(R, obtained vy adjoinig to the group Γ

(n/h) its

Atkin-Lehner involutions. See [38] or [105] for more details.

The subgroup G corresponding to a McKay-Thompson series T

(z) was con jec t u r ed expl i c-

itly by Conway and Norton for each g ∈ M. The su bgroup G is speciﬁed by giving the

integer N and a s u b set of Hall divisors of n/h. In fact, N arises as the least positive integer

156

such tha t



1 0

N 1



(z) = T

(z),

and n arises as the order o f g. Then, n divides N and the quotient h = N/n divides 24. In

fact, h

divides N. The subgroup G is given by

G =



a b

c d



∈ SL

(R) :



a b

c d



(z) = ζT

(z) for some ζ ∈ C with ζ

= 1



We give a few e xam ples to illustrate the situation.

Example 7.3.4. M has 5 conjugacy c la sses of elements of order 10 . They all ar i se from

the congruence subgroup with N = 10. We have h = 1, n = 10, n/h = 10 and the Hall

divisors of n/h form a group isomorphic to Z

× Z

. The ﬁve subgroups of this group are:

{1, 2, 5, 10}, {1, 2},

{1, 5}, {1, 1 0}, {1}.

There are 5 corresponding subgroup s G o f t h e n or m a li zer of Γ

(10) giving the 5 conjugacy

classes of M or order 10.

Example 7.3.5. M has 6 conjugacy classes of elements of order 6. Five of them arise from

the congruence subgroup with N = 6. In this case, we have h = 1, n = 6, n/h = 6 and the

Hall divisors of n/h form a gr ou p isomorphic to Z

×Z

. The ﬁ ve subgroups of this group

are:

{1, 2, 3, 6}, {1, 2},

{1, 3}, {1, 6 }, {1}.

There are 5 corresponding subgroups G of the normal i zer of Γ

(6). The remaining conjugacy

class of elements of order 6 arises from the congruence subgroup with N = 18. In this case,

we have h = 3, n = 6, n/h = 2, The Hall divisors of n/h form a group iso m or p h i c to

. The unit subgroup {1} is the one giving the required subgroup G of the n or m al i zer of

(18).

Example 7.3.6. M has 2 conjugacy classes of elements of order 78. They arise from the

congruence subgroup with N = 78. In this case we have h = 1, n = 78, n/h = 78 and the

Hall divisors of n/h form a group isomorphic to Z

× Z

. The two subgrou p s which

we require here are:

{1, 2, 3, 13, 6, 26, 39, 78},

{1, 6, 26, 39}.

These give rise to the two required subgroups G of the normalizer of Γ

(78).

These exampl es show some of the subtlety of the correspondences between subgroups of

(R) and conjugacy classes of M. The full cor r espondence can be found in [38, Table 2].

157

7.4 Replicable functions

A conjecture in [38] that played an important role in proving the main Conway-Norton

conjecture involves the replication formulae. Conway and Norton init ia l ly thought of the

Hauptmoduls T

as being intimately con n ect ed with M; if so, then the grou p structure of

M should somehow directly relate diﬀerent T

. Considering the power map g → g

leads

to th e following.

It was well known classically the following

Proposition 7.4.1. The function J(z) (equiv alently, j(z)) has the property that

K(z) = J(pz) + J





+ J



z+1



+ . . . + J



z+p−1



(7.13)

is a polynomial in J(z), for any prime p.

Proof. The proof i s straightforward, and is b as ed on the principle that the easiest way to

construct a function invariant with r espect to some group G is by averaging it over the

group:



g∈G

f(gx). Here f(x) is J(pz) and G is the modul ar group P SL

(Z), a n d we

average over ﬁnitely many cosets rather than inﬁnitely many elements.

First, writing Γ for P SL

(Z), n o t e that



p 0

0 1



Γ =



p 0

0 1



Γ ∪

p−1



i=0



1 i

0 p



Γ = {A ∈ GL

(Z) : det A = p} = S

. (7.14)

Here, we have used the fact that we know all the matrices in the group S

, according to

Example 7.1.8, for p prime.

Now, applying the Hecke operator T (p) to the modular form J(z) (with weight 2k = 0),

by equat i on (7.9) we have

T (p)J(z) = p

2k−1



ad=p, 0≤b<d



az + b





G∈S

J(Gz)



J(pz) + J





+ J



z+1



+ . . . + J



z+p−1





K(z).

Thus, K(z) = pT (p)J(z). Since J is a modular fo r m , Proposition 7.2.1 implies th at K(z) is

also a m odular form, hence a rational function on

Q(J(z))

P (J(z))

, by Theorem 6.5.2. Since the only

poles of J(z) are at the cusps, th e same applies t o K(z). This implies that the denominat o r

polynomial P (J(z)) must be trivial (recall that J(H) = C). Thus, K(z) is a polynomial in

J(z).

158

In particular, because K(z) is a modu la r invariant (it is co n st a nt on orbits of H/Γ, it must

satisfy K(z) = K( z + 1) (see also Proposition 6.2.2). Thus, K(z) must have a q-expansion

K(z) =



n∈Z

, wh er e q = e

2πiz

More gen er al l y, the same argument says th a t



ad=n, 0≤b<d



az + b



= Q

(J(z)), (7.15)

is a polynomial in J(z). In fact, Q

is the unique polynomial for which −q

−n

+ Q

(J(z))

has a q-e xp a n si on with only strictly positive powers of q (see Appendix A for more details).

For example, Q

(x) = x, Q

(x) = x

−2c

, Q

(x) = x

−3c

x−3c

, where J(z) =



In fact, thes e equations (7.15) can be rewritten into recursio n s such as a





+ a

, or

can be collected together into the remarkable remarkable identity originall y due to Zagier

[187], but discovered indepen d ently by Borcherds and ot h er s

−1



m≥1, n∈Z

(1 − p

)

= J( y) − J(z), (7.16)

with p = e

2πiy

, q = e

2πiz

and the powers c

are the coeﬃcients of t h e q-expansion of the

modular fu nction J(z). This is directly used in the proof of the Monstrous Moonshine

conjecture.

Conway and Norton conjectured in [38] tha t these formulae have an analogue for any

McKay-Thompson series T

. In particular, (7.15) becomes



ad=n, 0≤b<d



az + b



= Q

n,g



(z)



, (7.17)

where th e Q

n,g

plays t h e same role as Q

plays for J in (7.15). For example

(2z) + T





+ T



z+1



= T

(z)

− 2c

(1),

(3z) + T





+ T



z+1



+ T



z+2



= T

(z)

− 3c

(1)T

(z) − 3c

(2).

These are called the replication formulae. Again, these yield recursi ons like c

(4) = (c

(1)

−

(1))/2 + c

(3), or can be collected into the exp r essi on

−1

exp



−



k>0



m≥1, n∈Z

(mn)



= T

(y) − T

(z). (7.18)

D. Zagier (personal communication) has pointed that he was not the ﬁrst one who proved this ide ntity.

This kind of relations was already known by many other authors, and also appeared in [38]. In de ed , he

proved this formula and other similar relations in [187].

159

This looks a lot more complicated than (7.16), but we can glimpse the Taylor expansion

of log(1 − p

) there. In fact, taking g = 1, th e identity element of M, equation (7.18)

reduces to (7.16). Equation (7.1 8 ) is called the Borcherds’ identity, and we will give its

proof on Chapt er 8.

Axiomatising (7.17) leads t o Conway and Norton’s notion of replicable fun ct i on .

Deﬁnition 7.4.2. Let f be any function on H of the form f(z) = q

−1



n≥1

, and

write f

(1)

= f and a

(1)

= a

. Let X

= X

(f) the unique monic polynomial of degree n

such tha t the q-expansion of −q

−n

+ X

(f(z)) has only positive powers of q. Use



ad=n, 0≤b<d

(a)



az + b



= X



(1)

(z)



to deﬁne recursively each f

(s)

, for s ≥ 2. In each f

(s)

has a q- expansion of the form

(s)

(z) = q

−1



n≥1

(s)

—that is, n o fractional power s of q arise—, then we call f a

replicable function.

The r ead e r can see Appendix A to get a bett er idea of these functions. Just as a matter of

information, we have the following characterization of replicable functions:

Proposition 7.4.3. Let f be a function of the form q

−1



n≥1

, and deﬁne X

(f)

as in Deﬁnition 7.4.2. Then, f is replicable if and only if H

m,n

= H

r,s

holds whenever

mn = rs and (m, n) = (r, s).

Proof. The proof is not diﬃcult. Taking the expansion form of the functions X

(f). If f

is repl i cable, with replicates f

(s)

= q

−1



n≥1

(s)

, then

m,n



s|(m,n)

(s)

mn/s

(by Theorem 7.2.2), and the H

m,n

= H

r,s

property man i fest s. The converse follows in a

similar way.

Equation (7.17) conjectures that the McKay-Thompson series are replicable. In parti cu l ar ,

we have (T

)

(s)

= T

. Cummins and Norton [44] proved that the Hauptmodul of any genus

0 mod u l ar group of moonshine-type is replicable, provided it s coeﬃcients are rational. In-

cidentally, if the coeﬃcients a

are irrational, the n Deﬁnition 7.4.2 should be modiﬁed to

include Galois au t om or phisms (see [40]).

Any function f obeyin g the replicable functions (7.17 ) will also obey modular equations,

i. e., a certain type of 2-variable polinomial identities sat i sﬁed by f(x) and f(nx). The

160

simplest examples come from t h e exponential and cosine function s: note that for any n ≥ 1,

exp(nx) = (exp(x))

and cos(nx) = T

(cos(x)), where T

is a Tchebychev polynomi al . It

was known clasically that j (and hence J) satisfy a modular equation, for example, the

invariant quantity in (7.13). Note that this property of J depends crucially on it being a

Hauptmodul. Conversely, does the existen ce of modular equ at i ons force the Hauptmodul

property? Unfort unately not; both the exponential and cosine trivially obey modular

equations for each n (use Tchebychev polynomials for cos(nz)). However, we have the

following remarkable fact [124]: The only functions f(z) = q

−1

+ a

q + a

+ . . . which

obey modular equations for all n ar e J and the ‘modular ﬁcti ons’ q

−1

and and q

−1

±q (which

are essentially exp, cos, and sin). More generally, Cummings [43] p r oved the following.

Theorem 7.4.4. A function B(q) = q

−1



n≥1

which obeys a modular equation for

all n ≡ 1 (mod N), will either be of t he form B(q) = q

−1

+ a

q, or will be a Hauptmodul

for a modular group of Moonshine-type.

The converse is also true [43]. This theorem is the desired al geb r a ic interpretation of the

genus 0 property. The denominator id entity argum ent in Chapte r 8 will tells us that each

obeys a modular equation for each n ≡ 1 modulo t he order of g, so Theorem 7.4.4 then

would con cl u de the proof of Monstrous Moonshine. Moreover, Norton has conjectured

Conjecture 7.4.5. Any replicable function with rational coeﬃcients is either a Hauptmodul

for a genus 0 modular group of moonshine-type, or is one of the ‘modular ﬁctions’ f(z) =

−1

= e

−2πiz

, f(z) = q

−1

+ q = 2 cos(2πz), f(z) = q

−1

− q = −2i sin(2πz).

This con ject u r e seems diﬃcult and is still open.

7.5 The replication formulae

Let g be and elem e nt of the Monster group M, and consider

(z) =



n∈Z

tr(g|V

♮

, wit h q = e

2πiz

the McKay-Thompson seri es of g. Let G be the genus 0 sub g ro u p of SL

(R) associated to

the con ju gacy class of g by Conway and Nort on . Let

′

(z) :

H/G → CP

be t he canonical isomorph i sm of Riemann surfaces obtained from the action of G on the

upper half-plane H. The n , the Moonshine conjecture asserts that T

′

(z) = T

(z).

Conway and Nort on noticed that the coeﬃcients of T

′

(z) satisfy certain recurrence formulas,

called replication formulae. Let

(z)

′

= q

−1



n>1

, wit h q = e

2πiz

161

Then, the replica t io n formulas express certain coeﬃcients c

in terms of sm al l er coeﬃcients

, related either to g or g

. In the case g = 1, such replication formulas for th e coeﬃcients

of J(z) had been obta i n ed by Mahle r [?]. In the case of arbitrary g ∈ M, the following

replication formulae were proved by Koike [118]. We write c

(n) to be the coeﬃcie nt of q

in T

′

(z), tha t is

(z)

′

= q

−1



n>1

(n)q

, wit h q = e

2πiz

We then have four replication formulas:

(4k) = c

(2k + 1) +

(k)

−

(k) +

k−1



j=1

(j)c

(2k − j); (7.19)

(4k + 1) = c

(2k + 3) − c

(2)c

(2k) +

(2k)

(2k) +

(k + 1)

−

(k + 1) +



j=1

(j)c

(2k + 2 − j) +

k−1



j=1

(j)c

(4k − 4j) +

2k−1



j=1

(−1)

(j)c

(4k − j); (7.20)

(4k + 2) = c

(2k + 2) +



j=1

(j)c

(2k + 1 − j); (7.21)

(4k + 3) = c

(2k + 4) − c

(2)c

(2k + 1) −

(2k + 1)

(2k + 1) +

k+1



j=1

(j)c

(2k + 3 − j) +



j=1

(j)c

(4k + 2 − 4j) +



j=1

(−1)

(j)c

(4k + 2 − j). (7.22)

We note in particular that

(4) = c

(3) +

(1)

−

(1), (7 .2 3)

but t h e second replication formula with k = 1 gives simply

(5) = c

(5).

The idea for obtaining such replication for mulae is explained in [38, Section 8] . These

replication formulae can be used to express c

(n), for all n, in terms onl y of c

(1), c

(2),

162

(3), c

(5), for various elements h ∈ M, which are powers of g.

This fact gave Borcherds his strategy for proving the Moonsh i n e conjecture. If the coeﬃ-

cients of t he McKay-Thompson series T

(z) could be shown t o satisfy th e same replicati on

formulae, and if their coeﬃcients c

(1), c

(2), c

(3) and c

(5) agree with those of T

′

(z), then

it would follow that T

′

(z) = T

(z). In fact, Borcherds was able to obtain such replicat i on

formulae for T

(z) by me an s of the theor y of inﬁn i t e dimensional Lie algebras, as we shall

explain in th e next chapter.

163

Chapter 8

Borcherds ’ proof of Conway-Norton

conjecture

Borcherds’ proof of Moonshine conjecture makes use of the properties of a Lie alge b r a

called the Monster Lie algebra [10]. This is an exampl e of what is known as gener al iz ed

Kac-Moody algebras, or Borcherds algebras [9, 11]. We shall ﬁrst descri be the properties

of Borcherds algebras and, subsequently concentrate on the Monste r Lie algebra.

8.1 Borcherds Lie algebras

Deﬁnition 8.1.1. A Lie algebra g over R is called a Borcherds algebra if it satisﬁes the

following axioms:

(i) g =



i∈Z

has a Z-grad i ng such that dim g

is ﬁnite, for all i = 0 (dim g

not need to

be ﬁ n i t e) .

(ii) There exists a linear map ω : g → g such that

• ω

= 1, the identity map on g;

• ω(g

) = g

−i

, for all i ∈ Z;

• ω = −1 on g

(iii) g has an invariant bilinear form ·, · : g × g → R, such that

• x, y = 0, if x ∈ g

, y ∈ g

and i + j = 0;

• ωx, ωy = x, y, for all x, y ∈ g;

• −x, ωx > 0, if x ∈ g

, i = 0, x = 0.

These axioms imply that g

is abelian and that the scalar product ·, ·

: g × g → R,

deﬁned by

x, y

= −x, ωy

is positive deﬁnite on g

, for all i = 0. This ·, ·

is called the contravariant bilinear form

on g. We shall now give some examples of Borcherds algebras.

165

Example 8.1 .2. Let a = [a

], i, j ∈ I, be a symmetr i c mat r i x wit h a

∈ R. The index set

I need not nec essar i l y be to ﬁnite —we assume it is either ﬁnite or countably inﬁnite—.

Thus, our matrix a may be an inﬁnite matrix. We assume that this matrix satisﬁes the

conditions

• a

≤ 0, if i = j;

• if a

> 0, then 2

∈ Z, for all j ∈ I.

There is a Borcherds algebra g associated to the matrix a which is deﬁned by gener at or s

and rel at i o n s as follows. g is generate d by elements

, f

, h

for i, j ∈ I,

subject to the rela t io n s

• [e

, f

] = h

• [h

, h

kℓ

] = 0,

• [h

, e

] = δ

• [h

, f

] = −δ

• if a

> 0 and i = j, then

(ad e

)

= 0, (ad f

)

= 0, where n = 1 − 2

• if a

≤ 0, a

≤ 0 and a

= 0, then

, e

] = 0, [f

, f

] = 0.

This Li e algebra g can be graded by the condition

deg e

= n

, d eg f

= −n

for some n

∈ Z

. Ther e is an involution ω : g → g satisfying

ω(e

) = −f

, ω(f

) = −e

There is also an invariant bilinear for m on g uniquely determined by the condit io n e

, f

 =

1, for all i ∈ I. We wri t e h

= h

. Then, e

, f

 = h

and

h

, h

 = [e

, f

], h

 = e

, [f

, h

] = e

, a

 = a

for all i = j. Thus, h

, h

 = a

, for all i, j ∈ I.

We therefore see that th e Lie algebra g satisﬁes the axioms of a Borcherds algebra (Deﬁ-

nition 8.1. 1). It is called the universal Borcherds algebra associated with the matrix [a

166

The grading on g can be chosen in many ways, depending on the choice of the positive

integers n

We next observe that any sy m m et r i sable Kac-Moody algebra over R (recall Sect io n 4.3)

gives rise to a universal Borcherds algebra. For that, let g be the Kac-Moody algebra over

R with sym m et r i sab l e generalized Car t an matrix A = [A

]. Thus, there exists a diagonal

matrix

D =













with ea ch d

∈ Z

, such that the matrix DA i s symmetric. Let a = [a

] be given by

, for all i, j.

Then we have a

= a

and a

= d

. Thus a

≤ 0 if i = j and a

is a positive integer.

Also,

= A

∈ Z.

Thus, the symmetric matrix [a

] satisﬁes the conditions needed to construct a Borcherds

algebra, and the universal Borcherds algebra with symmetric matri x [a

] coincides with

the sub a lg eb r a of the Kac-Moody algebra g obtained by generators and relations prior to

the adjunction of the commutative algebra of outer derivations (see Deﬁnition 4.3.3). The

diﬀerence between a symmetrisable Kac-Moody algebr a and a uni versal Borcherds algebra

is tha t , in a Borcherds algebra:

1. The index se t I may be countably inﬁn it e rather that ﬁnite;

2. The a

’s may not be positive and need not lie in Z;

3. 2

is onl y assumed to lie in Z when a

> 0.

The center of a u n i versal Borcherds algebr a g lies in th e abelian subalgebra generated by

the elements h

and contains a ll h

with i = j. In fact, it can be se en that h

= 0 unless

the i-t h and j-th colu m n s of a are identical. If we factor out an ideal I of g which lies in

the center, then g/I retains the struct u r e of a Borcherds algebra. If we then adjoi n to g/I

an abelian Lie algebr a a of outer derivations to g ive the Lie algebra

∗

= (g/I) · a,

where a ⊆ (g

∗

)

and [e

, x] ∈ Re

, [f

, x] ∈ Rf

, for all x ∈ a, then g

∗

retains the structure

of a Borcherds algebra.

The converse is also true. G iven any Borcherds algebra g, there is a unique universal

Borcherds al gebra g

and a homomorphi sm f : g

→ g (not necessarily unique), such that

167

• Ker f lies in the center of g

• Im f is an ideal of g,

• g is the semi d i r ect product of Im f with a commutative Lie al geb r a of outer derivati ons

lying in the 0-graded component of g and pr eser vi ng all subspaces Re

and Rf

The hom o m or p h i sm f preserves the grading, involution and bi l inear form.

8.2 The Borcherds character formula

Let g be a universal Borcherds algebra. Recall from Section 4.3 that the root lattice Q of g

is the free abelian group with basis r

, for i ∈ I, with symmetric bilinear form Q ×Q → R,

given by

, r

) → r

, r

 = a

The basi s elements r

are call ed the simple roots. We have a grading

g =



α∈Q

determined by e

∈ g

, f

∈ g

−r

. We have seen in Sect i on 4.3 that an element α ∈ Q

is called a root of g if α = 0 and g

= 0. The root α is ca ll ed a positive root if α

is a sum of si m p l e roots. For any root α, ei t h er α or −α is positive. Let Φ = Φ

∪ Φ

−

be the set of roots of g. We say that α ∈ Φ is real if α, α > 0, and imaginary if α, α ≤ 0.

Remember also from Section 3.5 that the Weyl group W o f g is the group of isometries of

the r oot l at t i ce Q generated by the reﬂections w

corresponding to the simple real roots.

We have from (3.5)

) = r

− 2

r

, r



r

, r



= r

− 2

We recall that 2

∈ Z, since a

> 0. Let h be the abelian suba lg eb r a o f g generated by

the elements h

, for all i, j ∈ I. We have a map Q → h under which r

maps to h

, which

is a homomorphism of abelian grou p s and preserves the scalar product. However, this map

need n ot be injective.

If g is any Borcherds algebra, the root system and Weyl group of g i s deﬁned to be that of

the corr esponding universal Borcherds algebra.

We now introduce certain irreducible modules for a Borcherds algebra. Recal l from Section

4.3 that if g is a ﬁnite dimensional simple Lie algebra over C, the irredu ci b l e ﬁnite dimen-

sional g-mod u l es are in 1-1 correspondence with do m i n ant integral weights. Remember

that a weight λ ∈ h

∗

is dominant and integral if, and on l y if, λ(h

) ≥ 0 and λ(h

) ∈ Z, for

all i ∈ I. The weight λ arises as the h i ghest weight of this module, where λ, µ ∈ h

∗

satisfy

168

λ ≻ µ if and on l y if λ − µ is a sum of simple roots.

Now these ﬁnite di m ensional irreducible g-m odules are also in 1-1 corresponde n ce with

antidominant integral weights, i. e ., weights λ ∈ h

∗

satisfying λ(h

) ≤ 0 and λ(h

) ∈ Z, for

all i ∈ I. For there is a un i q u e lowest weight for the module, and th i s is antidominant and

integral. In the case of Borcherds algebras, i t is most convenient to consider lowest weight

modules rather than hig h est weight modules.

Recall that if g is a ﬁ n i t e dimen si onal sim p l e Lie algebr a and λ is an antidominant inte-

gral weight, the corresponding ﬁnite dimensional irreducibl e lowest weight module M

has

character g iven by the Weyl’s character formula (4.16)

(char M



α∈Φ

(1 − e

) =



w∈W

ǫ(w)w(e

λ+ρ

where ρ = −



(this is an alternative way of writing (4.16), making use of th e denomi-

nator i d entity (4.17)).

Next, suppose that g is a symmetrisable Kac-Moody algebra and λ ∈ h

∗

is a weight sat-

isfying λ(h

) ≤ 0, λ(h

) ∈ Z, for all i ∈ I. Then, g has a corresponding irreducible lowest

module M

whose character is given by Kac’ s character formula (4.18)

(char M



α∈Φ

(1 − e

)

mult α



w∈W

ǫ(w)w(e

λ+ρ

), (8.1)

where ρ ∈ h

∗

is any element satisfying ρ(h

) = −1, for all i ∈ I. This time the sum and

product may be inﬁ n i t e.

Finally suppose that g is a Borcherds algebra. Let λ ∈ Q ⊗ R be a weight satisfying

• λ, r

 ≤ 0, for all i ∈ I;

• 2

λ,r



r



∈ Z, for all i for which r

, r

 > 0.

Then, there is a corresponding irreducible lowest weight module M

whose character is

given by Borcherds character formula

(char M



α∈Φ

(1 − e

)

mult α



w∈W

ǫ(w)w



λ+ρ



α∈Q

ǫ(α)e



, (8.2)

where ρ ∈ Q ⊗ R sati sﬁ e s

ρ, r

 = −

r

, r

, for all i with r

, r

 > 0,

and ǫ(α) = (−1)

if α ∈ Q is a sum of k orthogonal simpl e imagi n ar y r oots all orthogonal

to λ, and ǫ(α) = 0 otherwise. Th i s formula reduces to Kac’s character formula (8.1) in

169

the case of a symmetrisable Kac-Moody algebra, since in this case there are no simpl e

imaginary roots and so



α∈Q

ǫ(α)e

= 1.

In the special case λ = 0, the modul e M

is the tri vi al 1-dimensional module and Borcherds’

character for mula becomes



α∈Φ

(1 − e

)

mult α



w∈W

ǫ(w)w





α∈Q

ǫ(α)e



. (8.3)

This is called Borcherds denominator identity. It general i zes Kac’s deno m inator id entity

(4.19), which was itself a generalization of Weyl’s den om i n a t or identity (4.17). As we

shall see, Borcherds’ denominator identity plays a key role in the proof of the Moonshine

conjecture.

8.3 The monster Lie algebra

We now consider an example of a Borcherds algebra, called the Monster Lie algebra [1 0 ],

which is our second main object of study, and plays an important role in the proof of the

Moonshine conjecture.

We start with the Monster vertex algebra V

♮

, constructed in Chapter 5. Re cal l that V

♮

has

a conformal vector of central charge 24. We will r ep l ace it with a vertex operator algeb r a of

central charge 26. Let Π be the lattice of rank 2 whose symmetric bilinear scalar product

Π × Π → Z is given by

b

, b

 = 0, b

, b

 = −1, b

, b

 = 0,

where b

, b

is a basis of Π. Thus we have

mb

+ nb

, m

′

+ n

′

 = −mn

′

− m

′

There is a vertex a lg eb r a V

associated with the lattice Π as in Section 4.5, and V

has a

conformal vector of central charge 2.

The tensor product V

♮

⊗V

also has the structure of a vertex operator algebra. This vertex

algebra has a conformal vector

ω ⊗ 1 + 1 ⊗ ω

of central charge 26, where ω and ω

are conformal vectors of V

♮

and V

, respectively.

Both vertex algebras V

♮

, V

have symmetric bilinear forms, and these deﬁne a symmetric

170

bilinear form on V

♮

⊗ V

. We deﬁne the subspaces

= {v ∈ V

♮

⊗ V

: L

(v) = v, L

(v) = 0 for i ≥ 1}; (8.4)

= {v ∈ V

♮

⊗ V

: L

(v) = 0, L

(v) = 0 for i ≥ 1}. (8.5)

Now recall from Remark 4.1.3 that the quoti ent (V

♮

⊗ V

)/T (V

♮

⊗ V

) has the structure

of a Lie algebra. The space P

/T (V

♮

⊗V

) ∩P

can be identiﬁed with a L i e subal geb r a of

♮

⊗ V

)/T (V

♮

⊗ V

). In fact, we have T P

⊆ P

and

T P

= T (V

♮

⊗ V

) ∩ P

Thus, P

/DP

has the structure of a Lie algebra ( see [10]).

The symmetric bilinear form on V

♮

⊗ V

induces such a form on P

, and T P

lies in the

radical of this form. Thus, we obtain a symmetric bilinear form on the Lie algeb r a P

/T P

Let M be the quotient of t h i s Lie algebra by the rad i ca l of the bilinear form, that is,

M =

/T P

Rad·, ·

In fact, M is itself a Li e algebra.

Deﬁnition 8.3.1. The Lie algebra M is called the Monster Lie algebra.

Now the vertex algebra V

has a grading by elements of th e lattice Π and this induces

gradings of V

♮

⊗ V

, and then of its subquotient M by elements of Π. We write

M =



m,n∈Z

(m,n)

, (8.6)

where (m, n) is the graded component corresponding to mb

+ nb

∈ Π. It was realized by

Borcherds that a theorem fr om string theory, known as the no-ghost theorem, applies to

this si tuation.

8.3.1 The no-ghost theorem

By the remarkable impor t an c e of applyi n g this th eor em in th e proof of Moonshine conjec-

ture, we mention a slight version of the no-ghost theorem (used by Borcherds in [12]). The

idea of using the no-ghost theorem to prove results about Kac-Moody algebras appeared

in Frenkel’s paper [65], which als o contains a proof of the no-ghost theorem. The original

proof of Goddard and Thor n [76] works for the cases we need with only trivial modi ﬁ ca -

tions. For conve n i en ce we give a sketch of their proof.

171

Theorem 8.3.2 (The no-ghost t h eo re m ) . Suppose that V is a vector space with a non-

singular bilinear form ·, ·, and suppose that V is acted on by the Virasoro algebra of

Deﬁnition 4.2.2 in such a way that

• the adjoin t of L

is L

−i

• the central element of the Virasoro algebra acts as multiplication by 24,

• any vector of V is a sum of ei genvectors of L

with nonnegative integral eige nvalues,

• and all the eigenspaces of L

are ﬁnite dimensional.

We let V

i−1

the subspace of V on which L

with has eigenvalue i. Assume that V is acted

on by a group G which preserves all this structure. We let V

be the vertex algebra of the

two dimensional even lattice Π (so that V

is Π-graded, has a bilinear form ·, ·, and is

acted on by the Virasoro algebra). We let P

be the subspace of V ⊗V

as deﬁned in (8.4),

and we let P

be the subspace of P

of degree α ∈ Π. All these spaces inherit an action of

G from the action of G on V and the trivial action of G on V

and R

Then, the quotient of P

by the nullspace of its bilinear form is naturally isomorphic, as a

G-module with an invariant bilinear form, to



−α,α/2

if α = 0

⊕ R

if α = 0

The nam e ‘no-ghost theorem’ comes from the fact that in the original statement of the the-

orem in [76], V was part of the underlying vector space of the vertex algebra of a positive

deﬁnite lattice, so the inner product on V

i−1

was positive deﬁnite, and thus, P

had no

ghosts ( i . e., vectors of neg at i ve norm) for α = 0.

We give a sketch of proof taken from [12] and [76]. Fix some nonzero α ∈ Π and some

norm 0 vector w ∈ Π, with α, w = 0. We use the following operators. We have an action

of the Virasoro algebra on V ⊗ V

generated by its conformal vector. The operat or s L

the Vira sor o algebra satisfy the relations (4.13)

, L

] = (i − j)L

i+j



i + 1



i+j,0

26,

and t h e adjoint of L

is L

−i

(the 2 6 comes from the 24 in (4. 13 ) plus the dimension of Π).

We d eﬁ ne operators K

, for i ∈ Z, by K

= v

i−1

, where v is the element e

−w

−2

of the vertex

algebra o f Π, and e

is an element of the group ring R[Π], corresponding to w ∈ Π, and

−w

is its inverse. These operators satisfy the relations

, K

] = −jK

i+j

, [ K

, K

] = 0,

since w has norm 0 and the ad joi nt of K

is K

−i

We deﬁne the following subspaces of V ⊗ V

172

• H is the subspace of V ⊗ V

, of degree α ∈ Π. H

is its subspace of vectors h with

(h) = h.

• P is the subsp ace of H of all vectors h wit h L

(h) = 0, for all i > 0. P

= H

∩ P .

• S, the space of spurious vectors, is the sub sp ac e of H of vectors perpendicular to P .

= H

∩ S.

• N = S ∩ P is the r adical of the bilinear form of P , and N

= H

∩ N.

• T , the transverse space, is the subspace of P ann i h i l at ed by all the operators K

, for

i > 0, and T

= H

∩ T .

• K is the space generat ed by the action of the operator s K

, i > 0.

• V e

is the subspace V ⊗ e

of H.

We have the following inclusions of subspaces of H:

S P K

V e

and we construct the isomorphism from V

−α,α/2

to P

/N ∩P

by zigzagging up and down

this diagram; more p re cis el y we show th at V e

and T are both isomorphic to K modulo

its nullspace, and then we show that T

is isomorphic to P

modulo its nullspace P

∩ N.

In fact, the no-ghost theorem follows immediately from the next sequence of lemmas [12]:

Lemma 8.3.3. If f is a vector of nonzero norm in T , then the vectors of the form

. . . K

. . . (f)

for all sequences of integers with 0 > m

≥ m

≥ . . ., 0 > n

≥ n

≥ . . ., are linearly

independent and span a space invariant under the operators K

and L

on which the bilinear

form is nonsingular.

Lemma 8.3.4. Th bilinear form on T is nonsingular, and K is the direct sum of T an d

the nullspace of K.

Lemma 8.3.5. V e

is naturally isomorphic to T .

Lemma 8.3.6. The associative algebra generated by the elements L

, for i < 0, is generated

by elements mapping S

into S.

Lemma 8.3.7. P

is the direct sum of T

and N

173

Recall from (8.6) that

M =



m,n∈Z

(m,n)

where ( m, n) is the graded component corresponding to mb

+ nb

∈ Π. Applying the

no-ghost theorem t o this vertex algebra, this theorem asserts th at for α ∈ Π,

is isomo r p h i c to V

♮

−α,α/2

, i f α = 0,

where V

♮

= {v ∈ V

♮

: L

(v) = (i + 1)v}. Let α = mb

+ nb

∈ Π. Since α, α = −2mn,

we then have

(m,n)

∼

♮

, i f α = (0, 0).

The no- gh o st theorem also asserts in this situation that

(0,0)

∼

The graded components of the monster Lie algebra M can ther efor e be shown in the fol-

lowing t able:

0 0 0 0 0 V

♮

0 0 0 0 0 V

♮

0 0 0 0 0 V

♮

0 0 0 V

♮

−1

0 V

♮

··· 0 0 0 0 R

0 0 0 0 ···

♮

0 V

♮

−1

0 0 0

♮

0 0 0 0 0

♮

0 0 0 0 0

♮

0 0 0 0 0

Table 8.1: Grad ed compon ents of the Monster Lie algeb r a M.

The grou p ring R[Π] of the lattice Π has an involution deﬁned by

→ (−1)

α,α/2

−α

, for α ∈ Π.

This gives rise to an involution on the vertex algebra V

= S(

−

) ⊗ R[Π]. This in turn,

gives rise to an involution on the vertex algebra V

♮

⊗ V

, which acts trivially on V

♮

. This

involution acts on the subquotient M of V

♮

⊗ V

, giving a map ω : M → M such that

174

• ω

= 1,

• ωM

(m,n)

= M

(−m,−n)

• ω = −1 on M

(0,0)

• ωx, ωy = x, y;

where · , · is the invariant b i l i n ear form on M. Moreover, the contravariant form

x, y

= −x, ωy, for x, y ∈ M,

is positive d eﬁ nite on M

(m,n)

, for all (m, n) = (0, 0).

We can give M a Z-grad ing by means of the formula

deg M

(m,n)

= 2m + n.

The Z-g ra d ed compone nts are:

··· −5 −4 −3 −2 −1 0 1 2 3 4 5 ···

··· V

♮

⊕ V

♮

0 V

♮

−1

♮

−1

0 V

♮

⊕ V

♮

···

Table 8.2: Z-gr a d ed components of the Monster Lie al geb r a M.

Hence, M satisﬁes the axioms for a Borcherds algebra, having the necessary Z-grading,

involution, and invariant bilinear form giving rise to a positive deﬁnite contravari ant form

on non-zero graded components.

Let Q be th e root lattic e of the Borcherds algebra M, and let h be the Cartan subalgebra

of M. Then , h = M

(0,0)

= R

and we have a map Q → h as in Section 8.1, under which

each simple root r

∈ Q maps to h

∈ h, and preserving the scalar product. The elements

of h may be written in the form mb

+ nb

, for m, n ∈ Z. The elements of h which a ri se as

images of simple roots in Q are those with (m, n) equal to

(1, −1), (1, 1) , (1, 2), (1, 3), (1, 4), . . .

Thus, M has i n ﬁ n i t el y may simple root s (note that they are not linearly independent).

Since

mb

+ nb

, mb

+ nb

 = −2mn,

we see that (1, −1) gives a real simple r oot and that all th e other simple roots (1, n), for

n ≥ 1, are imaginary.

175

We have pointed out in Section 8.1 that the map Q → h need not be injective, and in

the present si t u at i on it is far from injective. Thus, there can be several simple roots in Q

mapping t o the same element b

+nb

of h. The number of simp le roots mapping to a given

element b

+ nb

is called the multiplicity of (1, n). This multiplicity is

dim M

(1,n)

= dim V

♮

= c

, (8.7)

where c

is the coeﬃcient of q

in th e expansion of the normalized Hauptmodul (1.4)

J(z) = q

−1



n≥1

, wit h q = e

2πiz

Thus,

• (1, −1) has multiplicity 1,

• (1, 1) has multiplicity 196,884,

• (1, 2) has multiplicity 21,493,760;

• . . .

and the sum of th e simpl e roo t spaces in M is isomorphic to the Moonshi n e module V

♮

Hence, the Monster Lie algebra M contains within it the Monster vertex algebra V

♮

as the

sum of its root spaces corresponding to the simple roots.

The symmetri c matrix [a

] corresponding to the Borcherds algebra M is thus a countable

matrix with many repeated rows and columns (see Table 8.3). It has the following form:

Since M has onl y one simple real root, its Weyl group W has or d er 2. Any root of M maps

to an element mb

+ nb

∈ h such that M

(0,0)

= 0 and (m, n) = (0, 0). The multiplicity of

(m, n) is then

dim M

(m,n)

= dim V

♮

= c

8.4 Denominator identities

In this section we will describe Borcherds’ denom i n a to r identity and twisted denominator

identity for the monster Lie algebra M. First , we obtain the Borcherds’ deno m i n at or iden-

tity for th e monster Lie algebra, supposing only we know completely the simple roots of M.

Remember from Sect i on 8.1 the Borcherds identity (8.3)



α∈Φ

(1 − e

)

mult α



w∈W

ǫ(w)w





α∈Q

ǫ(α)e



176

(1, −1) (1, 1) ··· (1, 1) (1, 2) ··· (1, 2) (1, 3) ···

(1, −1) 2 0 ··· 0 -1 ··· -1 -2 ···

(1, 1) 0 -2 ··· -2 -3 ··· -3 -4 ···

(1, 2) -1 -3 ··· -3 -4 ··· - 4 -5 ···

(1, 3) -2 -4 ··· -4 -5 ··· - 5 -6 ···

Table 8.3: Cartan matrix for the monster Lie algebra M.

where ρ ∈ Q ⊗ R is any vector satisfying

ρ, r

 = −

r

, r

, ∀i ∈ I.

Consider Q → h the hom o m or p hism described in Section 8.1, map p i n g si m p l e roots in Q

to h. By abuse of language, we will also call these elements in h of roots. In the case of the

monster Lie algebra M, simple root s in h are the elem ents mb

+ nb

∈ h, with m, n ∈ Z

and mn > 0 or mn = −1. If there are k roots in Q mapping to the same root in h, we will

say that this root in h has multiplicity k. Also, we identify the root mb

+ nb

with the

pair (m, n) ∈ Z

. So, we know that the simple roots of M are

(1, −1), (1, 1) , (1, 2), (1, 3), (1, 4), . . .

and we could take ρ = (−1, 0), because

(−1, 0), (1, n) = n,

(1, n), (1, n) = −2n, for all n,

so that ρ, r

 = −

r

, r

 for each simple root r

. Also, we know that root (m, n) has

multiplicity exactly c

Let p = e

(1,0)

and q = e

(0,1)

. We have, e

= e

−(1,0)

= p

−1

, so left-hand side of Borcherds’

identity ( 8 .3) is

−1



m>0, n∈Z

(1 − p

)

177

(because e

(m,n)

= e

m(1,0)+n(0,1)

= (e

(1,0)

)

(0,1)

)

= p

). Remember al so that for α ∈ Q,

ǫ(α) = (−1)

, when α is the sum of k imaginary orthogonal simple root s , and ǫ(α) = 0 in

other ca se.

For the monster Lie algebra M, there are no two imaginary orthogonal simple roots, because

(1, m), (1, n) = −m − n < 0, for all m, n. Thus, the elements α ∈ Q contributing to sum



ǫ(α)e

are α = 0 with ǫ(α) = 1 and all the imaginary simple roots (1, n) ∈ h, n ∈ Z

Since there are precisely c

of these roots in Q (mapping to (1, n)) and all of them have

ǫ(α) = −1, we have



α∈Q

ǫ(α)e

= 1 −



n>0

. (8.8)

Also, |W | = 2 and W = {1, s}, where s(p) = q and s(q) = p. Thus, right-h an d side of

Borcherds’ identity is



w∈W

ǫ(w)w





α∈Q

ǫ(α)e





w∈W

ǫ(w)w



−1



1 −



n>0





= p

−1



1 −



n>0



− q

−1



1 −



n>0





−1

−



n>0



−



−1

−



n>0



= j(p) − j(q). (8.9)

Combining (8.8) and (8.9), the denominator identity for the monster Lie algebra M estab-

lishes Zagier’s id entity (7.16)

−1



m>0, n∈Z

(1 − p

)

= j(p) − j(q). (8.10)

In fact, this identity were ﬁrst proved by Borcherds in terms of the M structure, and then

were used for prove that the simple roots of M are (1, − 1) , (1, 1), (1, 2), (1, 3), . . .

8.5 The twisted denominator identity

In order to complete the proof of Moonshine conjecture, we shall need a more general form

of id entity (8.10), named th e twisted denominator identity. To explain it, we will give an

outline of proof of Zagier’s identity (8.10), and then we wil l generalize it.

178

Deﬁnition 8.5.1. Let U be a ﬁnite-dimensional r ea l vector space, with a graded decompo-

sition U =



α∈L

, for some lattice L. We deﬁne the graded dimension of U as the element

in R[L] given by

gr dim U =



α∈L

(dim U

where R[L] is the group algebra of L with basis e

, for α ∈ L.

Let ∧

U, ∧

U, . . . be the exterior powers of U, that is

∧

U = {k-linear alternate forms ω : U × . . . × U → R},

and ∧

U = R. We have a simple for mula for the alternate sum



k≥0

(−1)

gr dim ∧

given by



k≥0

(−1)

gr dim ∧

U =



α∈L

(1 − e

)

dim U

. (8.11)

Note th at the right-hand side can also be writ t en as exp



−



k>0



α∈L

(dim U

kα



, since

exp





α∈L

(dim U

)



k>0

−

kα



= exp





α∈L

(dim U

) log(1 − e

)





α∈L

(1 − e

)

dim U

Suppose now that U is a G-module, for some ﬁnite group G, and that G a ct s on each

graded component U

Deﬁnition 8.5.2. The graded character of U, as the map gr char U : G → R[L] given by

g −→



tr(g|U

Then, t h e alternate sum



k≥0

(−1)

gr dim ∧

U is given by th e map

g −→ exp



−



k>0



tr(g

kα



Observe that when g = 1 , this map r ed uces to the alternat e sum in (8.11 ) . If U is an

inﬁnite dimensi on a l vector space and U =



α∈L

is a graded decomposition of U, such

179

that each component U

is ﬁni te dimension al , then the formulas are still valid (changing

the sum s for inﬁnite sums).

Suppose now that g is a Borcherds al gebra with triangular decomposi t i on

g = n

⊕ h ⊕ n

−

where n



α∈Φ

and n



α∈Φ

−

(note that n

can be inﬁnite dimens io n al , but

each g

must be ﬁnit e dimension al ) . Consider the exterior powers

∧

U, ∧

U, . . .

We have a sequence

. . .

−→ ∧

−→ 0

with h o m ol og y groups

Ker d

Im d

, H

Ker d

Im d

, H

Ker d

Im d

, . . .

Observe that ∧

and H

are graded vector spaces with ﬁnite dimensional graded

components. By deﬁnition of the homology groups H

, we have



k≥0

(−1)

gr dim ∧



k≥0

(−1)

gr dim H

Garland and Lepowsky [73] p r oved that for Kac-Moody algebras (and also it is veriﬁed for

Borcherds algebras), H

can be identiﬁed with a subspace of ∧

as follows:







(∧

)

if α + ρ, α + ρ = ρ, ρ

0 in other case

and this holds for each α in the root lattice of g.

Now, we specialize for the case g = M, the monster Lie algebra. The formula (8.11) gives



k≥0

(−1)

gr dim ∧



(m,n), m>0

(1 − p

)

where p = e

(1,0)

, q = e

(0,1)

and M



α∈Φ

180

We also compute the alternate sum



k≥0

(−1)

gr dim H

. We have, gr dim ∧

= e

i. e., ∧

= R is 1-dimension a l with weight e

. Since 0 + ρ, 0 + ρ = ρ, ρ, we also have

gr dim H

= e

. Next we have

gr dim ∧

= gr dim M



(m,n), m>0

Now we consider gr dim H

. We have ρ = (−1, 0). Let α = (m, n) be a root with

m > 0. Then α + ρ, α + ρ = −2(m − 1)n and ρ, ρ = 0, so we get α + ρ, α + ρ = ρ, ρ

if and only if m = 1 or n = 0. Since M

has no roots with n = 0 and the roots (m, n) wi t h

m = 1 are precisely the sim p l e roots, we obtain

gr dim H



(1,n)

= p





n∈Z



Now, the weights of ∧

are sums of two distinct weights in M

. Since all weights of

are of the form (m, n) with m ≥ 1 then there are no weights (m, n) in ∧

with

m = 1. However, there are some weights (m, n) wit h n = 0. These are precisely o f the

form (1 , −1) + (m − 1, 1), wh e re m − 1 ≥ 1. It follows that

gr dim H



m≥2

m−1

Note that for ∧

the weights are sums of three distinct weights of M

. None has the

form (m, n) with m = 1 or n = 0. Then, H

= 0. Simi l ar l y, H

= 0 for all k ≥ 3.

Thus we have



k≥0

(−1)

gr dim H

= e

− p



n∈Z



m≥2

m−1

= e

+ p



m≥1

− p



n∈Z

= e

+ p





m∈Z

− p

−1



− p





n∈Z



= p



j(p) − j(q)



Therefore, we have derived the denominator identity (8.10)



(m,n), m>0

(1 − p

)

= p



j(p) − j(q)



181

In order to obtain the twisted denominator identity, we consi d e r M

as an M-module for

the Mon st er group M. Then, each root space (M

)

is an M-modu le. From that we have



k≥0

(−1)

gr dim ∧



k≥0

(−1)

gr dim ∧

(observe that each sub sp a ce (H

)

is also an M-m odule). Moreover, the left-hand side



(−1)

gr dim ∧

is the map

g −→ exp



−



k>0



α∈Φ

tr(g

|(M

)

kα



, (8.12)

for g ∈ M. If we replace all dimensions for characters in the formulas above involving

, H

and H

, we get that the su m



(−1)

gr char H

is the map

g −→ p





n∈Z

tr(g|V

♮

−



n∈Z

tr(g|V

♮



. ( 8. 13 )

Comparing (8.12) and (8.13 ) we deduce that

−1

exp



−



k>0



(m,n), m>0

tr(g

♮





n∈Z

tr(g|V

♮

−



n∈Z

tr(g|V

♮

. (8.14)

This is what we have called the twisted denominator identity for the monster L i e algebra M.

Observe that if we wr i t e c

(n) = tr(g|V

♮

) for the coeﬃcient of q

in the graded character



n∈Z

tr(g|V

♮

, then equation (8.14) is just

−1

exp



−



k>0



(m,n), m>0

(mn)p





n∈Z

(n)p

−



n∈Z

(n)q

or simply

−1

exp



−



k>0



(m,n), m>0

(mn)p



= T

(y) − T

(z),

where y = e

2πip

, z = e

2πiq

. That is, exactly the form of equat i on (7.18).

8.6 Replication formulae aga in, an d proof ’s end

The twisted denominator identity for the monster Lie al g eb r a (8.14) is just what is needed

to obtain the replicat i on formulae (7.19)-(7.22). By comparing the coeﬃcients of p

and p

182

in this identity and applying some elementary algebra, Borcherds d er i ved the replication

formulae. We omit the proof of this fact here, because the abundance of calculations. The

complete derivation of the replication formulae is made in Appendix A.

This is alm ost suﬃcient for the proof of the Conway-Norton conjecture. In order to complete

the proof, i t rema ins to show that the coeﬃcients c

(1), c

(2), c

(3) and c

(5) of the

McKay-Thompson series T

(z), agree with those of T

′

(z). These coeﬃcients for T

′

(z) were

completely obtained by Conway and Norton in [38]. In order to obtain the coeﬃcients for

the graded characters T

(z), it is suﬃcient to know how the modules V

, V

and V

the Monster M, with dimensions c

, c

and c

, respectively, decompose into irreducib l e

modules. Observe that the only irreducible characters of M less or equal than c

= dim V

are

, χ

(see Table 1.2), t hus these are the only possibl e irreducible components of V

, V

and

. Borcherds was able to p roof th at

dim V

= χ

+ χ

dim V

= χ

+ χ

dim V

= 2χ

+ 2χ

+ χ

dim V

= 4χ

+ 5χ

+ 3χ

+ 2χ

+ χ

, (8.15)

where as u su a l, χ

, χ

, . . . , χ

are the ﬁrst 7 irreduc ible characters of M (denoted by d

in Chapter 1 ) . This was proved by ﬁnding 7 elements g

, g

, . . . , g

of M for which the

7 × 7-matrix [χ

)] is non-singular and by showing that the above equations (8.15) hold

when eval u a t ed at each g

. They must then hold for all g ∈ M.

We mention in conclu si o n , th a t the proof of the Conway-Norton conjecture for th e Monster

is by no means the sole achievement of Borcherds’ work in [12]. Other spor a d i c simple

groups are also discussed, including the Baby Monster B, th e Conway group Co

, th e

Fischer group F i

′

, the Harada-Norton group HN, the Held group He, and the Mathieu

group M

, and denominator identities for all th ese groups are also obtained. Thus, the

topic of Monstrous Moonshine is by no me an s conﬁned to the Monster M.

183

Chapter 9

Concluding Remarks

We give in this chapter a quick sketch of further development s and conjectures. As can be

seen, Moonshine is an area where it is much easier to conjecture than to prove.

9.1 Orbifolds

In string t h eor y, the m o st tractable way to introduce singularities is by quotienting (‘gaug-

ing’) by a ﬁnite group. This construction plays a fundamental role for CFT and vertex

operator al geb r a s; it is the physics underlying what Norton calls generalized Moonshine.

This is where ﬁnite group theory touches CFT. Let M be a manifold and G a ﬁnite grou p

of symmet r i es of M. The set M/G of G-orbits inherits a topology from M, a n d forms a

manifold-like space called an orbifold. Fixed points become conical singularities. For ex-

ample, {± 1} acts on M = R by multiplication. The orbifold R/{±1} can be identiﬁed with

the interval [0 , ∞). Th e ﬁxed point at x = 0 becomes a sing u l ar point on the orbifold, that

is, a point where locally the orbifold d oes not look like some open n-ball. Orbifolds were

introduced into geometry in the 1950’s as spaces wit h certain kin d of singular i t ie s. They

were introduced into string theory in [49], which greatly increased the class of background

space-times in which the string could live and still be amenable to calculation. This section

brieﬂy sketches the corresponding construction for CFT; our purpose is to moti vate some

generalization of Moonshine conjecture.

About a third of the McKay-Thompson series T

have some negative coeﬃcients. We shall

see Borcherds int er p r et them as dimensions of superspaces (which come with signs). In

the important announcement [155], on a par with [38], Norton proposed that, although

(−1/z) will not usually be another McKay-Thompson series, it will always have non-

negative integer q-coeﬃcients, and th ese can be interpreted as ordin ar y dimension s . In

the process, he extended the g → T

assignment to commuting pairs (g, h) ∈ M × M. In

particular

Conjecture 9.1.1 (Norton). To each such pair g, h ∈ M, with gh = hg, we have a function

(g,h)

(z), such that

)

(z) = αN



g, h;

az+b

cz+d



, for all



a b

c d



∈ SL

(Z), (9.1)

185

for some root of unity α (of order dividing 24, and depending on g, h, a, b, c, d). N

(g,h)

(z) is

either constant, or generates the modular functions for a genus 0 subgroup of SL

(Z) con-

taining some Γ(N) (but otherwise not necessarily of moonshine-type). Constant N

(g,h)

(z)

arise when all elements of the form g

(with (a, b) = 1) are ‘non-Fricke’. Each N

(g,h)

(z)

has a q

1/N

-expansion for that N; the coeﬃcients of this expansion (not necessarily integers)

are characters ev aluated at h of some central extension of the centralizer C

(g). Simulta-

neous conjugation of g, h leaves the Norton series unchanged: N

(aga

−1

,aha

−1

)

(z) = N

(g,h)

(z).

We call N

(g,h)

(z) the Norton series associated to (g, h). An element g ∈ M is called Fricke

if the group G

contains an element sending 0 to i∞ —the id e ntity 1 i s Fricke, as are 120

of the 171 G

—. For example, when g, h

∼

×C

and g, h, gh are all in class 2A, then

(g,h)

(z) = q

−1/2

−492q

1/2

−22590q

3/2

+. . ., while N

(g,g)

(z) = q

−1/2

+4372q

1/2

−96256q+. . ..

The McKay-Thompson series are recovered by taking g = 1: N

(1,g)

= T

. This action (9. 1 )

of SL

(Z) is related to its natural action on the fundamental group Z

of the torus, as well

as a natural action of the braid group, a s we shall see in the next s ect i on . Norton arrived

at hi s conjecture empirically, by studying the data of Queen (Section 9.3).

The basic tool we have for approaching Moonshine conjectures is the theory of vertex op-

erator algebras, so we need to understand Norton’s suggestion from that point of view. For

reasons of space, we limit our discussion to V

♮

, bu t it generalizes. Given any automor-

phism g ∈ Aut(V

♮

), we can deﬁne g-twisted modules in a straightforward way [51]. Th en

for each g ∈ M, there is a un i que g-twisted module, call it V

♮

(g),. More generally, given

any a u t om o r p h i sm h ∈ Aut(V

♮

) commuting with g, h will yield an automorphism of V

♮

(g),

so we can perform Thompson’s graded characters (1.9) a n d deﬁne

(g,h)

(z) = q

−c/24

♮

(g)

. (9.2)

These Z

(g,h)

can be thou ght of as the building b l ocks of the graded dimensions of various

eigenspaces in V

♮

(g); for example if h has order m, then the subspace of V

♮

(g) ﬁxed by

automorphism h will have graded dimension m

−1



i=1

(g,h

)

. In the case of the Monster

considered here, we have Z

(g,h)

= N

(g,h)

At t h e l evel of algebr a, this orbifold theory is analogous to t h e construction of twisted

aﬃne algebras from nontwisted ones. At the level of mod u l ar forms, it involves twists an d

shifts much like how θ

(z) =



(−1)

and θ

(z) =



(n+1/2)

/2 are obtained from

(z). Far from an esoteric technical development, orb ifo lds are central to the whole theory,

and a crucial aspect of Moonshine. The important p aper [51] proves that, whenever the

subgroup g, h generated by g and h is cyclic, then N

(g,h)

will be a Hauptmod u l satisfying

(9.1). One way this will happen of course is whenever the orders of g and h are coprime.

Extending [51] to all commuting pairs ( g, h) is one of the most pressing tasks in Moonshine.

At least some aspects of orbifolds are more tractable in the subfactor framework (see for

example [55] , so further investigations in that direction should be fruitful. This orbifold

construction is the same as was use d to construct V

♮

from V

; V

♮

is the sum of the ι-

invariant subspace V

♮

of V

with the ι-invariant subspac e V

♮

−

of the unique −1-twisted

186

module for V

, where ι ∈ Aut(Λ

) is some involution. The graded dimensions of V

♮

are

−1

(Z(±1, 1) + Z(±1, ι)), respectively, and these sum to J.

The orbifold construction is al so involved in an interesting reformulation of the Hau p t m odul

property, due to Tuite [177]. Assume the following

Conjecture 9.1.2 (Uni qu eness of V

♮

). V

♮

is the only vertex operator algebra with graded

dimension J.

Tuite argues from th i s that, for each g ∈ M, T

will be a Haup t m odul if and onl y if

the only orbifolds of V

♮

are V

and V

♮

itself. In [104], this analysis is extended to

some of Norton’s series N

(g,h)

, where the subgroup g, h is not cyclic (thus going beyond

[51]), although again assuming the uniqueness conjecture. Recently [24], [25] (and other i s

preparation), Carn a h an has outlined an approach to the generalized mooshine conjecture

by usin g Borcherds’ products.

9.2 Why the Monster?

In the work [189], Zhu introduced a particular algebra with special features that provided

some results about t h e modularity of some functions arisi n g on vertex operator algebras.

The fact that M is associated with modular functions can be explained by it being the au-

tomorphism group of the Moonshine vertex operator algebra V

♮

and the foll owing theorem

Theorem 9.2.1 (Zhu’s Theorem). Suppose V is a C

-coﬁnite weakly rational VOA (see

[189] for the deﬁnitions), and let Φ(V ) be the ﬁnite set of irreducible V -modules. Then,

there is a representation π of SL

(Z) by complex matrices π(A) indexed by V -modules

M, N ∈ Φ(V ), such that the one-point functions

(z, v) = tr

o(v)q

−c/24

= q

c/24



n≥0

h+n

o(v)q

h+n

obey



az + b

cz + d

, v



= (cz + d)



N∈Φ(V )



a b

c d



(z, v),

for any v ∈ V obeying L

v = nv, for some n ∈ N.

What is so speci al about this gro u p M that these modular functions T

and N

(g,h)

should

be Hauptmod uls? Thi s is stil l open. One approach is due to Norton, a n d was ﬁrst stated

in [155]: the Monster is probably the largest (in a sense) group with the 6-tran sposition

property. Recall that a k-transposition group G is one generated by a conjugacy class K

of involut i on s , where the product gh of any two elements of K has order at most k. For

example, taking K to be the transpositions in the symmetric group G = S

, we ﬁnd t h a t

is 3-tr ansposition.

187

A transitive action of Γ = P SL

(Z) on a ﬁnite set X with one distingui sh ed point x

∈ X,

is equivalent to specifying a ﬁnite index subgroup Γ

of Γ. In particular, Γ

is the stab i l i zer

{g ∈ Γ : g·x

= x

} of x

, X can be identiﬁed with the cosets Γ

\Γ, and x

with the coset Γ

To su ch an action, we can associat e an interesting triangu la t io n of the closed surface Γ

called a (modular) quilt. The deﬁnition, origin al l y due to Norton and further developed

by Parker, Co nway, an d Hsu, is somewhat involved and will be avoided h er e (but you can

see [98]) . It is so-named because there is a polygonal ‘patch’ covering every cusp of Γ

\H,

and the closed surface is formed by sewin g together the patches along their edges ‘seams’

(Figure 9.1).

Figure 9.1: A ‘friendly’ process of compactifying and sewing a 4-p unctured pl an e .

There are a tot al of 2n triangles and n seams in the triangulation , where n is the index

188

|Γ

\H| = |X|. The boun d ar y of each patch has an even number of edge s, namely the

double of the correspondi n g cusp width. The familiar formula

γ =

−

∞

+ 1

for the genus γ of Γ

\H in term s of the index n an d the numbers n

of Γ

-orbits of ﬁxed

points of order i, can be interpreted in terms of the data of the quilt ( see [98]), and we ﬁnd

in particular that i f every patch of the quilt has at most six sides, then the genus will be 0

or 1, and genus 1 only exceptionally. In particular, we are interested in one class of these

Γ-actions (actuall y an SL

(Z)-action). For example, it is well known that the braid g ro u p

has pr es entation

σ

, σ

| σ

= σ

,

and center Z = (σ

)

 (see [5]). It is related to t h e modular group by

∼

P SL

(Z), B

/(σ

)



∼

(Z).

Fix a ﬁnite group G (we are most interested in the choice G = M). We can deﬁne a right

action of B

on tri p l es (g

, g

) ∈ G

, g

)σ

= (g

−1

, g

), (g

, g

)σ

= (g

, g

−1

, g

). (9.3)

We will be interested in this action on the subset of G

where all g

∈ G are involu-

tions. The action (9.3) is equi valent to a reduced version, where we replace (g

, g

) with

, g

) ∈ G

. Then (9.3) becomes

(g, h)σ

= (g, gh), (g, h)σ

= (gh

−1

, h). (9.4)

These B

actions come from specializations of th e Bu ra u a n d r educed Burau representa-

tions respectively [5], [114], and genera li ze to actions of B

on G

and G

n−1

. We can get

an acti on of SL

(Z) from the B

action (9.4) in two ways: either

(i) by restricting to commuting pairs (g, h); or

(ii) by identifying each pair (g, h) with all its conjugates (aga

−1

, aha

−1

Norton’s SL

(Z) action of (9.1) arise s from the B

action (9.4) when we perform both (i)

and (i i ) .

The quilt picture was designed for this SL

(Z) action. The po int of this construction is

that the number of sid es in each patch is determin ed by the order s of the correspond ing

elements g, h. If G is, say, a 6-tr a n sposition group ( su ch as the Monster), and we take the

involutions g

from 2A, then each patch will have ≤ 6 sides, and the corresponding genus

will be 0 (usually) or 1 (exceptionally). In this way we can relate the Monst er wi t h a genus

0 property.

Based on the actions (9.3) and (9.4), Norton anticipates some analogue of Moonshine

valid for noncommuting pairs. Alth ou g h they alway s seem to be modular functions, they

189

will no longer always be Haup t m oduls and their ﬁxing groups won’t always contain a

Γ(N). Conformal ﬁeld theory considerations ( ‘ h i gher genus orbifolds’) alluded in Section

9.1 suggest t h a t more natural should be for example quadruples (g, g

′

, h, h

′

) ∈ M

obeying

[g, h] = [h

′

, g

′

An important question is, how much does Monstrous Moonshine determine the Monster?

How much of the M structure can be deduced from, for example, McKay’s

Dynkin

diagram observation , and/or the (complete) replicability of the McKay-Thompson series

, and/or Norton’s conjectures 9.1.1, and/or Modu l ar Moonshine in Section 9.4 below?

A small start toward this is taken in [157], where some control on the subgroups of M

isomorphic to C

×C

(p prime) was obtained, using only the properties of t he series N

(g,h)

For related work, see [98].

9.3 Other ﬁnite groups: Mini-Moonshine

It is natural to ask about Moonshine for other grou p s. For example, the Hauptmodul for

(2)

looks like

(2)

(q) = q

+ 4, 372q + 96, 256q

+ 1, 240, 002q

+ . . . (9.5)

and we ﬁnd the relations 4, 372 = 4, 371+1, 96, 256 = 96, 255+1, 1, 240, 002 = 1, 139, 374+

4, 371 + 2 ·1, where 1, 4371, 96255, and 113 937 4 ar e all d im e n si on s of irreducible represen-

tations of the Baby Monster B. Thus we ﬁnd ‘Moonshine’ for B.

Of course any subgroup of M automatica l ly inher i t s Moonshin e by restriction, but this

is not at all interesting. A more accessible sporadi c is M

(see for example[39]). Most

constructions o f the Leech lattice start with M

, and most constructions of th e Monster

involve the Leech lattice. Thus we are led to the following natu ra l hierarchy of (most )

sporadics:

1. M

(from wh i ch we can get M

, M

); which leads to

2. Co

∼

2.Co

(from wh i ch we get HJ, HS, McL, Suz, Co

, Co

); which leads to

3. M (from whi ch we get He, Fi

, Fi

, Fi’

, HN, Th, B).

It can thus be argued that we coul d approach problems in Monstrous Moonshine by ﬁrst

addressing in order M

and Co

, which should be much simpler. Indeed, the full vertex

operator algebra orbifold t h e or y (the compl et e analogue of Section 9. 1 ) for M

has been

established in [52] (the relevant seri es Z

(g,h)

had already been constructed in [ 14 0]). The

orbifold theory for Co

though seems out of reach at present. Remarkab l y, that for the

Baby Monst e r B is much m o r e straightforward and has been worked out by H¨ohn [93].

190

Queen [159] established Moonshine for the following grou p s (all essentially centralizers of

elements of M): Co

, Th, 3.2. S u z, 2.HJ, HN, 2.A

, He, M

. In particular , to each ele-

ment g of these gr ou ps, there corresponds a series Q

(z) = q

−1



n≥0

(g)q

, which is a

Hauptmodul for some modular group of Moon s h i n e- type, and where each g → a

(g) is a

virtual character. For example, Q u ee n ’ s series Q

for Co

is the Hauptmodul (1.10) for the

genus 0 group Γ

(2). For Th, HN, He and M

it is a proper character. Other diﬀerences

with Monstrous Moonshine are that there can be a preferred nonzero value for the constant

term a

, and that although Γ

(N) will be a subgrou p of th e ﬁxing gr ou p, it will not neces-

sarily be normal. We will return to these results i n the next section, wher e we will see that

many seem to come ou t of the Moonshine for M. About half of. Queen’s Hauptmodul s

for Co

do not arise as a McKay-Thompson series for M. Norton conjecture 9.1.1 are a

reinterpretation and extension of Queen ’ s work.

Queen never reached B because of its size. However, the Moonshine (9.5) for B falls into

her and Nor t on ’ s scheme because equation (9.5) is the McKay-Thompson series associated

to class 2A of M, and the centralizer of an element in 2A is a double cover of B.

There can not be a vertex operator al geb r a V =



with graded dimension (9.5) and

automorphisms in B, because for example the B-module V

does not contain V

as a sub-

module. However, H¨ohn deepened t h e analogy between M and B by constructing a vertex

operator superalgebra V B

♮

of rank c = 23

, called the shorter Moonshine module, closely

related to V

♮

(see for example [94]). Its automorphism group is C

× B. Just as M is the

automorphism group of the Griess algebra B = V

♮

, so is B the automorphism group of the

algebra (V B

♮

)

. Just as V

♮

is associated to the L eech la t ti ce Λ

, so is V B

♮

associated to

the shorter Leech lattice O

, the unique 23-dim en si onal positive-deﬁnite self-dual lattice

with no vectors of l en g t h 2 or 1 (see for example [39]). The automorphism group of O

is C

× Co

. A si m i l ar theory has recently appeared for Co

in [58]. There has been no

interesting Moonshine for the remaining six sporadics (the pariahs J

, J

, Ru, O’N, Ly, J

There will be so m e sort of Moonshine for any group which is an automorphism group of

a vertex operator algebra (so this means any ﬁnite group [53]). Many ﬁnite groups of Lie

type should arise as automorphism groups of vertex operator algebras associated to aﬃne

algebras except deﬁned over ﬁ n i t e ﬁelds, but apparently all known examples of genus 0

Moonshine are l im i t ed to the groups involved with M.

Lattices are r el at e d to groups through their automorphism grou p s, which are always ﬁnite

for positive-deﬁnite lattices. The automorphism grou p Aut Λ

= Co

of the Leech lattice

has ord er abou t 8 × 10

, and is a central exten s io n by Z

of Conway’s simple group

. Several other sporadic groups are also involved in Co

, as we have seen. To each

automorphism α ∈ Co

, let θ

denote the theta function of the sublattice of Λ

ﬁxed

by α. Conway and Norton also associate with each automorphism α a certain function

(z) of the form



η(a

z) /



η(b

z) built out of the Dedekind ’ s η function (Example

1.4.2). Both θ

and η

are constant on each conjugacy class in Co

, of which there are

167. [38] remarks that t h e ratio θ

/η

always seems to equal some McKay-Thompson

191

series T

g(α)

. It turns out that this observat i on is not correct [126] . For each autom or p hism

α ∈ Co

, the subgroup of SL

(R) that ﬁxes θ

/η

is indeed always genus 0, but for exactly

15 conjugacy classes in Co

, θ

/η

is not the Hau p t m odul. Nevertheless, this construction

proved useful for establishi ng Moonshine for M

[140]. Similarly, one can ask this for the

root lattice, whose automorphism group is the Weyl group of t he Lie algebra E

(of

order 696,729,600). The automorphisms of the lattice E

that yield a Hauptmodul were

classiﬁed in [162]. On t h e other hand, Koike estab l i sh ed a Moonshine of this kind for the

groups P SL

), P SL

)

∼

Alt

and P SL

), of o r d er 168, 60 and 12, respectively

[119, 1 21 , 120, 122, 123].

9.4 Mod ul ar Moonsh ine

Consider an element g ∈ M. We expect from [ 15 5], [159], [51], that there is a Moonsh i n e for

the centralizer C

(g) of g in M, governed by the g-twisted module V

♮

(g). Unfortuna t ely,

♮

(g) is not usually itself a vertex operator algebra, so the analogy wit h M is not perfect.

Ryba and Borcherds [161], [19], [16] found it interesting that, for g ∈ M of prim e order p,

the Norton series N

(g,h)

can be transformed into a McKay-Thompson series (and has all the

associated n i ce properties) whenever h is p-regular (that is, h ha s order coprime to p). This

special behavior of p-regular elements suggested to him to look at modul ar representations.

The basics of modular representations and Brauer characters are discussed in suﬃcient

detail in [45].

A modular representation π of a group G is a repres entation deﬁned over a ﬁeld of positive

characteristic p dividing the order |G|. Such representations possess many special features.

For one thing, they are no longer completely redu ci b l e (so the role of irreducible modules

as direct summands will be replaced with their role as compo si t i on factors). For another,

the usual notion of character (the trace of representation matrices) loses its usefulness and

is replaced by the more subt l e Brauer character β( π): a complex-valued class function on

M which is onl y well-deﬁn ed on the p-regular elements of G. We have, for exampl e (see

[16], [19], [161]).

Theorem 9.4.1. Let g ∈ M be any element of prime order p, for any p dividing |M|.

Then, there is a vertex operator superalgebra

V =



n∈Z

deﬁned over the ﬁnite ﬁeld F

and acted on by the centralizer C

(g). If h ∈ C

(g) is p-regular, then the graded Brauer

character

(g,h)

(z) = q

−1



n∈Z

β(

)(h)q

equals the McKay-Thompson series T

(z). Moreover, for g belonging to any conjugacy

class in M except 2B, 3B, 5B, 7B, or 13B, this is in fact an ordinary vertex operator algebra

(that is the ‘odd’ part vanishes), while in the remaining cases the graded Brauer characters

of both the odd and e ven parts can separate ly be expressed using McKay-Thompson series.

192

By a vertex operator superalgebra, we m ea n t h er e is a Z

-grading into even and odd

subspaces, and for u, v both odd, th e com mutator in the locality axiom of Theorem 4.1.2 is

replaced by an anticommutator. In the proof, the superspaces arise as cohomology groups,

which naturally form an alternating sum. The centralizers C

(g) in the theorem ar e quite

nice; for examp le for g in classes 2A, 2B, 3A, 3B, 3C, 5A, 5B, 7A, 11A, r espectively,

these involve the sporadi c groups B, Co

, Fi’

, Suz, Th, HN, HJ, He, and M

. The

proof for p = 2 is not complete at the present time. The conjectures in [161] concerning

modular analogu es of the Griess algebra for several sporadics follow from Theorem 9. 4. 1.

Can these m odular

V vertex operator algebras be interpreted as a red u c t io n (mod p)

of (su per)algebras in characteristic 0? Also, what about elements g of composite order?

Borcherds has stated the following in [16]

Conjecture 9.4.2 (Borcherds). Choose any g ∈ M and let n denote its order. Then, there

is a

Z-graded superspace

V =



i∈(1/n)Z

over the ring of cyclotomic integers Z[e

2πi/n

It is often (but probably not always) a vertex operator superalgebra; in particular,

V is an

integral form of the Moonshine module V

♮

. Each

V carries a representati on of a central

extension of C

(g) by C

. Deﬁne the graded trace

(g,h)

(z) = q

−1



i∈

tr(h|

If g, h ∈ M commute and have coprime orders, then B

(g,h)

(z) = T

(z). If all q-coeﬃcient s

of T

are non-negative, then the ‘odd’ part of

V vanishes, and

V is the g-twisted module

♮

(g). If g has prime order p, then the reduction (mod p) of

V is the modular vertex

operator superalgebra

V of Theorem 9.4.1.

When we say

V is an integral form for V

♮

, we mean that

V has the same structure as a

vertex operator algebra, with everything deﬁned over Z, and tensoring it with C recovers

♮

. This remarkable conjecture, which tries to explain Theorem 9.4.1, i s completely open.

9.5 The geometry of Moonshine

Algebra is the mathematics of stru c t u r e, and so of course it has a profound relationship

with every area of mathematics. Therefore the trick for ﬁndin g possib le ﬁngerpr i nts of

Moonshine in, say, geometry is to look there for modular functions. That sear ch quickly

leads t o the elliptic genus.

For details see for example [9 5 ], [1 64 ] , or [170]. All manifold s here are compact, oriented

and diﬀerentiable. In Thom’s cobordism ring Ω, elements are equivalen ce classes of cobor-

dant manifolds, addition is connected sum , and multiplication is Cartesian produ ct . The

universal ell i p t i c genus φ(M) is a ring homomorphism from Q⊗Ω to the ring of power series

in q, which sends n-dimensional manifolds with spin connections to a weight

modular

193

form of Γ

(2) with integer coeﬃcients. Several variations and generalizations have been in-

troduced, e. g., the Witten genus assigns to spin manifolds with vanishing ﬁrst Pontryagin

class a weight

modular form of SL

(Z) wit h integer coeﬃcients.

We have noticed several deep relationships between elliptic genera Moonshine. For instance,

the i m portant rigidity property of the Witten genus with respect t o any compact Lie group

action on the manifold, is a consequence of the modula ri ty of the characters of aﬃne alge-

bras [135 ]. The elliptic genus of a manifold M has been interpreted as the gr ad ed dimension

of a vertex operat o r superalgebra constructed from M [169]. Seeming ly related to this, [21]

recovered the elliptic genus of a Calab i -Yau manifold X from the sh ea f of vertex algebras

in the chiral de Rham complex MSV[139] at t ached to X. Un e xpectedly, the ellip t i c genus

of even-dimensional projective spaces P

has non - n ega t ive coeﬃcients and in fact equal s

the graded dimension of some vertex al gebra [138]; this suggests interesting representation

theoretic questions in the spirit of Mon st r ous Moonsh ine. In physics, elliptic genera arise

as partition functions of N = 2 superconformal ﬁ el d theories [183]. Mason’s construc-

tions [140] associated to Moo n shine for the Mathieu group M

have been interpreted as

providing a geometric model (elliptic system) for elliptic cohomology Ell

∗

(BM24) of the

classifying space of M

[170], [54]. Th e Witten genus (norma li zed by η

) of the Milnor-

Kervaire manifold M

, an 8-dimensional manifold buil t from the E

diagram, equals j

1/3

[95] ( r eca ll (4.32)).

Another interesting fact is that a Borcherds algebr a can be associated with any even

Lorentzian lattice, and also with any Calabi-Yau manifold [92]. Of course it is a broad

enough class that almost all of them will be uninteresting; an intriguing approach to ide n -

tifying the interesting ones is by considering the so called automorphic product s [14, 15].

Hirzebruch’s ‘prize question’ [95] asks for the construction of a 2 4- d i m en si onal manifold

M with Witten genus J (aft er being normali zed by η

). We would like M to act on M

by diﬀeomorphisms, and the twisted Witten genera to be the McKay-Thompson series T

It would also be nice to associate Norton series N

(g,h)

to this Moonshine manifold. Con-

structing such a manifold is perhaps the remaining ‘Holy Grail’ of Monstrous Moonshine.

Hirzebruch’s question was partially answered by Mahowal d and Hopkins [137], who con-

structed a manifold with Witten genus J, but could n ot show that it would support an

eﬀective act i on of the Monster. Related work is by Aschbacher [1], who constructed several

actions of M on, for exam ple, 24-dimensional manifolds (but none of which could have

genus J), and Kultze [125 ] , who showed that the graded dimensions of the subspaces V

♮

of the Moonshine module are twisted

A-genera of the Milnor-Kervaire manifol d M

(the

A-genus is the specialization of elliptic genus to t he cusp i∞).

There has been a second conjectured relationsh i p between geometry and Monstrous Moon-

shine. Mirror symmetry s ays that most Calabi-Yau manifolds come in closely related

pairs. Consider a 1-parameter famil y X

of Calabi-Yau manifolds, with mirror X

∗

given

by the resolution of an orbifold X/G for G ﬁnite and abelian. Then the Hodge num-

194

bers h

1,1

(X) and h

2,1

∗

) will be equal, and more precisely the moduli space of (com -

plexiﬁed) K¨ahler structures on X will be locally isomet r i c to the moduli space of com-

plex structures on X. The ‘mirror map’ z(q), which can be d eﬁ ned using the Picard-

Fuchs equation [151], gives a canonical map between those moduli spaces. For example,

+ x

+ z

−1/4

= 0 is such a family of K3 surfaces (that is Calabi-Yau

2-folds), where G = C

× C

. Its mirror map is given by

z( q) = q − 104q

+ 6, 444q

− 311, 744q

+ 13, 018, 830q

− 493, 025, 760q

+ . . . . (9.6)

Lian and Yau [133] noticed th at the reciprocal

z(q)

of the mirror map in (9.6) equals the

McKay-Thompson series T

(z) + 104 for g in class 2A of M. After looking at several other

examples with simi la r conclusions, they proposed their Mirror-Moonshine conjecture: The

reciprocal 1/z of the mirror map z of a 1-parameter family of K3 surfaces with an orbifold

mirror will be a McKay-Thompson series (up to an addit i ve constant).

A counterexample (and more examples) are given in [180]. In particu l ar , although there are

relations between mirror symmetry and modular functions (see for exam p l e [89] and [92]),

there does not seem to be any special relation with the Monster. Doran [56] demystiﬁes

the Mirror-Moonshine ph enomenon by ﬁnding n eces sar y and suﬃcient cond i t io n s for 1/z

to be a modular function for a modular group commensurable with SL

(Z).

9.6 Moo ns hi ne and physics

The physical side (perturbative string theory, or equivalently conformal ﬁeld theory) of

Moonshine was noticed early on, and has pr ofoundly i n ﬂ u e n ced the development of Moon-

shine an d vertex operator algebras. This is a very rich subject , which we can only super-

ﬁcially touch on. The book [64], with its ext en si ve bibliography, provides an introductio n

but will be diﬃcult rea d i n g for many mathema t ic ia n s (as will this section). The treatme nt

in [70] is m o r e accessib le and shows how naturally vertex operator algebras arise from the

physics. This eﬀectiveness of physical interpretations is not magic; it m er el y tell s us that

many of our ﬁnite-dimensi on a l objects are seen much more clearly when studied thr ough

inﬁnite-dimensional structures. Of course Moonshine, which teaches us t o stud y the ﬁnit e

group M via its inﬁnite-dimension a l module V

♮

, ﬁts perfectly into this picture.

A confor mal ﬁeld theory i s a quantum ﬁel d theory on 2-dimensional space-time, whose sym-

metries incl u de the conformal transformations. In string theory the basi c objects are ﬁnite

curves —called strings— rather than points (particles), and the conformal ﬁeld theory lives

on the surface traced by the strings as they evolve (colliding and separating ) through time.

Each conformal ﬁel d theory is associated with a pair V

, V

of mutually commuting vertex

operator a lg eb r as , called its chiral algebras [4]. For example, strings living on a compact

Lie group manifold (the so-called Wess-Zumino-Witten model) will have chiral algebras

given by aﬃne algebra vertex operator algebras. The space H of stat es for the conformal

ﬁeld the or y carries a representation of V

⊗

, and many authors have (somewhat op t i -

mistically) concluded that the study of conformal ﬁeld theories redu ces to that of vertex

195

operator algebra r epresentation theory. Rational vertex operator algebras correspond to

the important cl ass of rational conformal ﬁeld theories, where H decomposes into a ﬁnite

sum ⊕M

⊗M

of irreducible modules. The Virasoro algebra of S ect i on (4.2) arises natu-

rally in conformal ﬁeld theory t h r ough inﬁnitesimal conformal transformations. Th e vertex

operator Y (φ, z), for the space-time parameter z = e

t+ix

, is the quantum ﬁeld which cre-

ates from the vacuum |0 ∈ H the state |φ ∈ H at time t = −∞ : |φ = lim

z→0

Y (φ, z)|0 .

In particular, Borcherds’ deﬁnition [8, 13] of vertex operator algebras can be interpreted

as an ax io m at i sat i on of the notion of chiral algebra in conform al ﬁeld theory, and for this

reason a lo n e is important.

In conformal ﬁel d theory, t he Hauptmodul property of Moonshine is hard to interpret, and

a less direct formulation, li ke that in [177], is needed. However, both the statement and

proof of Zhu’s Theorem are natural from the conformal ﬁeld theory framework (see [70]);

for example, the modularity of the series T

and N

(g,h)

are automatic in conformal ﬁeld

theory. This modularity arises in conformal ﬁeld t h e or y through the equivalen ce of the

Hamiltonian formulation, which d escr i bes concretely the graded spaces we take traces on

(and hence the coeﬃcients of our q-expansions), and the Feynman path formalism, which

interprets these graded traces as sections over moduli spaces (and hence makes modularity

manifest). Beautiful revi ews are sketched in [184], [183]. More explicitly, the Virasoro

action on moduli spaces discussed in section 4.2 of [71] gives rise to a syst em of partial dif-

ferential eq u at i ons (the Knizhnik-Zamolodchikov equat io n s) . According to conformal ﬁeld

theory, the vertex operator algebra characters will satisfy those eq u at i on s for a torus with

one puncture, an d their modularity (that is, Zhu’s Theorem) arises from the monodromy

of those equations.

Because V

♮

is so mathematically special, it may be expected that it corresponds to in-

teresting physics. Certainly it has bee n the subject of some speculati on. There wil l be a

c = 24 rational conformal ﬁeld theor y whose chiral algebra V

and state space H are both

♮

, while V

is trivial ( t his is possible because V

♮

is holomorphic). This conform al ﬁeld

theory is nicely d escr i bed in [48]; see also [50]. The Monster is the symmet r y of that con-

formal ﬁ e ld theory, but the Bimonster M ≀C

will be the symmetry of a ra t io n al conformal

ﬁeld theory with H = V

♮

⊗

♮

. The paper [41] ﬁnds a fami l y of D-branes for the latter

theory which are in one-to-one correspondence with the elements of M, and their ‘overlaps’

g||q

+c/24)

||h equal the McKay-Thompson series T

−1

. However, we still lack any

explanation a s to why a conformal ﬁeld theory involving V

♮

should yield interesting physics.

Almost every facet of Moonshine ﬁnds a natural formulation in conform al ﬁeld theory,

where it often was discovered ﬁrst . For example, the no-ghost theorem (Theorem 8.3.2)

of Brower-Goddard-Thorn was used to great eﬀect in [12] to understand the structu r e of

the Monster Lie algebra M. On a ﬁn i t e- d i m en si onal manifold M, the index of the Di r ac

operator D in the hea t kernel interpretation is a path integral in supersymmetric quantum

mechanics, that is, an integral over the free loop sp ace LM = {γ : S

→ M}; the string

196

theory version of this is that the in dex of the Dirac operator on LM should be an integral

over L(LM), that is over s m ooth maps of tori into M, and this is just th e elliptic genus,

and explains why it should be modular. The orbifold construction of [51] comes straight

from conformal ﬁeld theory (although const r uction of V

♮

in [66] predates conformal ﬁeld

theory orbifolds by a year and in fact inﬂuen ced their development in physics). That said,

the translation process from physics to mathematics of cours e is never easy; Borcherds’

deﬁnition [8] i s a prime example.

From this standpoint, what is most excit ing is what has not yet been fully exploited. String

theory tells us t h a t conformal ﬁeld th eor y can live on any su r face Σ. The vertex oper at or

algebras, including the geometri c vertex operator algebras of [99], capture conformal ﬁeld

theory in genus 0. Th e gr a d ed dimensions and traces considered above co n cer n conformal

ﬁeld theory quantities (conformal blocks) at genus 1: z → e

πiz maps H onto a cylinder,

and the tr ace identiﬁes the two ends. There are analogues of all this at higher genus [188 ]

(though the formulas can rapidly become awkward). For example, the graded dimension

of the V

♮

conformal ﬁeld theory in genus 2 is computed i n [178], and involves for instance

Siegel theta functions. The orbifold theory in Section 9.1 is genus 1: each ‘sector’ (g, h)

corresponds to a homomorphism from the fundamental group Z

of the tor u s into the orb-

ifold group G (for example G = M); g and h are t h e targets of the two generator s of Z

and

hence must commute. More generally, the sectors will correspond to each homomorph i sm

ϕ : π

(Σ) → G, and to each we will get a high er genus trace Z(ϕ), wh i ch will be a fu n ct i on

on the Teichm¨uller space T

(generalizing the up per half-plane H for genus 1) . The action

of SL

(Z) on the N

(g,h)

generalizes to the action of the mapping class group on π

(Σ) and

. See for examp le [6] for s om e thoughts in this direction . Recently, Witten [186] has also

related the Monster with 3- d i m en si onal gravity in black holes, although the paper is still

abundant i n conjectures.

9.7 Conclusion

There are diﬀerent basic aspects to Mon st r ou s Moonshine: (i) why mod u l ar i ty enters at all;

(ii) why in particular we have genus 0; and (iii) what it has t o do with the Monster. Today,

we understand (i) best. There will be a Moonshine-like relations between any (subgroup of

the) automorphism group of any ratio n al vertex operator algebra, and th e characters χ

and the same can be expected to hold of the orbifold characters Z in Section 9.1.

To prove the genus 0 property of the McKay-Thompson series T

, we needed recursions

obtained one way or another from the Monster L i e algebra M, and from these we apply

Theorem 7.4.4. These r ecu r si o n s are very special , but so presumably is the 0 genus prop-

erty. The suggestion of [28], though, is th at we may be able to considerably simplify this

part of the argument.

Every group known to have ri ch Moonshine properties is contained in the Monster. To what

extent can we deri ve M from Monstrous Moonshine? The understanding of this seeming l y

central r ol e of M is the poorest of those three aspects.

197

The central role that vertex operator algebras play in our current understanding of Moon-

shine should be clear from this review (that is basically Zhu’s Theorem). The excellent

review [54] makes this point even m or e forceful l y. However, it can be (and has been) ques-

tioned whether the full and diﬃ cu l t machinery of vertex operator algebras is really needed

to understand this; that is, whether we really have isola t ed the key conjunction of prop-

erties needed for Moonshin e to arise. Conformal ﬁeld theory has been an invaluable guide

thus far, but perhaps we are a little too steeped in its lore.

In particular, what it is really needed is a second in d e pendent proof of the Moonshine con-

jectures. One t em p t i n g possi b i l i ty is the heat kernel; its general role in modularity concerns

is emphasized in [109] , and is also th e central ingredient in the Knizhnik-Zamolo dchikov

equations for aﬃne algebras [96]. A heat kern el probably plays an analogous role in the

Knizhnik-Zamolodchikov equations associated to V

♮

, but does it relate to the genus 0

property? Another pos si b i li ty is the braid group B

, whose ﬁngerprints are all over the

mathematics and physics of Moonshine.

Moonshine (in its more general sense) is a relation between algebra and number theory,

and its impact on algebra ha s been dramatic (for example: vertex operator algeb r a s, V

♮

Borcherds-Kac-Moody algebras). Its impact on numb er t h eo r y has been far less so. This

may merely be a temporary acci d ent due to the backgrounds of m os t researchers (i n cl u d i ng

the mathematical physicists) working to date in the area. Gannon [71] has suggested t h a t

the most exciting prospects for the future of Moonshine ar e in the direction of number

theory. Hints of this fut u r e can be found in for example [15], [47], [57], [90], [150], [185]

and [18]. Other recent advances and discussions are [14 4 ] , [63], [37], [146], and [143].

198

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206

Appendix A

Proo f o f t he Repl ica t io n Formulae

This part is a continuation of Section 7.5. Our purpose is to give a detailed proof of the

recursion formulas ( 7. 19 ) - ( 7. 22 ) .

A.1 Faber polynomials

The Faber polynomials [60] originated in approximation theory in 1903 and are central to

the theory of replicable functions. We deﬁne them in a formal way. The reader interested

in anal y t ica l aspect s of these polynomials can consult [60] or sectio n 3 and 12 of [145 ] .

Let f : H → C be a function having q-expansion

f(z) =



n≥0

where we take q = e

2πiz

, for z ∈ H, the upper half-plane. Throug h out, we interpret

derivatives of f with respect to q. We initially assume that the coeﬃcients H

of f are

in C and we choose the con s t ant term to be zero. For each n ∈ Z

, th e re exists a uni q u e

monic polynomial P

in f , such that

(f) =

+ O(q) as q → 0,

(or equivalently, P

(f) ≡ q

−n

(mod qZ[q])). In fact, P

= P

(f) depends on the coeﬃcients

of f, but we denote i t simply by P

when there is no confusio n . The polynomial P

is called

the n-th Faber polynomial associ at ed with f. It can be shown that the Faber polynomials

are given by the generating series

′

(q)

z − f(q)



n≥0

(z)q

with P

(z) = 1, P

(z) = z, P

(z) = z

−2H

, P

(z) = z

−3H

, and more generally:

(z) = det(zI − A

207

where







(n − 2)H

n−3

n−4

n−5

. . . 1

(n − 1)H

n−2

n−3

n−4

. . . H

n−1

n−2

n−3

. . . H







It is useful to note that the Faber poly n om i al s satisfy a Newton type recurrence relation

of the form

n+1

(z) = zP

(z) −

n−1



k=1

n−k

(z) − (n + 1)H

, (A.1)

for all n ≥ 1.

Another useful way to see the Faber polynomials according to Norton [ 15 4 ] is the following.

Consider f with the q-expa n si on

f(z) =



n≥0

, q = e

2πiz

We deﬁne some coeﬃcients H

m,n

by the formula

F (y, z) = log



f(y) − f(z)



= log(p

−1

+ q

−1

) −



m,n≥1

m,n

(the bivarial transformation of f), wher e p = e

2πiy

. Then X

(f) =

−n



m,n

the coeﬃcient o f p

in th e expansion

−log p −log



f(y) − f(z)



so that it is a polynomial in f. In fact, X

(f) =

(f), and we can see X

as a (non-

monic) n-th Fabe r polynom i al associ at ed to f.

Example A.1.1. If f(z) is a modular form of weight 2k on SL

(Z), then the Hecke

operators T

, n ≥ 1 act on f as

(f)(z) = n

k−1



ad=n, 0≤b<d



az + b



for all n ≥ 1. See Chapter 6 and S ect io n 7.1, or the ﬁrst ﬁve chapter of [18 1 ] (especially

Zagier’s article) for background details. When k = 0 and f (z) is the j-function, we have

(j)(z) =



ad=n, 0≤b<d



az + b



208

The generators of SL

(Z) permute the lin ear fractional transformations in the sum, hence

(j) is invaria nt under SL

(Z). Since T

(j) has no poles in the upper half-plane H, it

follows t h at it is a polynomial in j (see Section 7.1 ) . We ﬁnd that,

(f)(z) =

+ O(q) as q → 0,

for all n ≥ 1, and so T

(j) =

(j). Thus the Hecke operator of the j-function are

examples of Faber polynom i al s.

A.2 Replicable functions

Let f be a function having q-expansion

f(z) =



n≥0

where q = e

2πiz

. Such a function is called replicable if ther e exist a sequence {f

(s)

}

s∈Z

functions f

(s)

: H → C, such that for all n ≥ 1, the expression

(f) =



ad=n, 0≤b<d

(a)



az + b



. (A.2)

The fun ct i on f

(s)

introduced above is called the s-th replication power of f.

Thus, we can think this polynomial P

(f) as the action of a generalized Hecke oper at or .

Following Norton [154], let us consider the coeﬃcients {H

m,n

}

m,n≥1

introduced in previous

section by

(f) =

−n



m≥1

m,n

, (A.3)

for n ≥ 1 (note in particu la r that H

n,1

= H

), an d the slightly modiﬁed ones

m,n

= (m + n)H

m,n

It follows from equation (A.1) that this h

m,n

are given recursively by

m,n

= (m + n)H

m+n−1

m−1



i=1

n−1



j=1

i+j−1

m−i,n−j

A useful characterization for replicable functions proved by Norton in [15 4 ] is th e following:

Theorem A.2.1. The function f is replicable if, and only if, H

m,n

= H

r,s

for all positive

integers m, n, r, s satisfying mn = rs and (m, n) = (r, s).

209

Let H

(s)

be the coeﬃci ent of q

in th e s-th replication power of f, that is

(s)

(z) =



n≥0

(s)

for all n, s ≥ 1 (i n p a r t ic u l ar , H

(1)

= H

). According t o Pr oposition 7.4.3 and Theorem

7.2.2, equation (A.2) can be written as

m,n



s|(m,n)

(s)

mn/s

. (A.4)

Applying the M¨obius inversion formula to equation, then we have

(s)

= s



d|s

µ(d)H

dsn,

, (A.5)

where µ is the M¨obius function.

Another result obt a ined by Norton [154] is:

Theorem A.2.2. Suppose that f is replicable. For any k ∈ Z

, as s ranges over divisors

of k, the following are equivalent:

1. f

(s)

is replicable.

2. The bivarial transform of f

(s)

is the generating function of H

(s)

m,n

3. In addition to condition (1), the t-th replication power of f

(s)

is f

(st)

Other r esu l t s about replicable functions can be found in [124] and [147].

A.3 Deduction of the replication formula e

Let f (z) be given by the series

f(z) =



n≥1

where q = e

2πiz

. D eﬁ ne X

(f) as in (A.3) (observe in particular that X

(f) =



n,1

= f (z). We will write X

= f .

We have already mentioned that X

is the coeﬃcient of p

in the expansion of log p

−1

−

log



f(y) − f(z)



, where p = e

2πiy

. Writin g

log p

−1

−



n≥1

= log



f(y) − f(z)



210

it foll ows that

−1

exp



−



n≥1



= p

−1



n≥1

− q

−1

−



n≥1

exp



−



n≥1



= 1 + p



− q

−1

−



n≥1





n≥1

n+1

. (A.6)

Using the Taylor expansion exp



−







k≥0



−





, and comparing the

coeﬃcients of p

, p

and p

in (A.6 ) , we obtain



− 2X

+ X





− 2X

+ f



; (A.7)



− 6X

+ 6X

f − f



; (A.8)



− 24X

+ 24X

f + 12X

− 12X

+ f



. (A.9)

Consider f

(s)



n≥1

(s)

, the replication powers of f. Also, deﬁne the linear operator

, n ≥ 1, such that for any series of the form



ℓ∈Z

ℓ

, we have





ℓ∈Z

ℓ





= n



ℓ∈Z

nℓ

ℓ

Then, Koike [118] proved the following formulas called 2-plication and 4-plication, respec-

tively:

(f) =



+ f

(2)

(2z)



; (A.10)

(f) =



+ f

(2)

(2z) + f

(4)

(4z)



. (A.11)

In fact , both these for mulas can be obtained from the recursion equation (A.4) as follows.

From (A.4 ) we have

n,2



(2)

n/2

, n ≡ 0 (mod 2);

, n ≡ 1 (m od 2);

and

n,4











(2)

(4)

n/4

, n ≡ 0 (mod 4);

(2)

, n ≡ 2 (mod 4);

, n ≡ 1, 3 (mod 4).

211

Hence,



+ f

(2)

(2z)





n≥1





n≥1

(2)



−2



n≥1



n≥1

(2)

−2



n≥1

n,2

= X

(f),

and



+ f

(2)

(2z) + f

(4)

(4z)





n≥1



n≥1

(2)







n≥1

(4)



−4



n≥1



n≥1

(2)



n≥1

(4)

−4



n≥1

n,4

= X

(f).

Using t h ese Koike’s formulas, from (A.7) and (A.10) we obtain



− f|

− f

(2)

(2z)



; (A.12)

and from (A.7)-( A. 11) :



−

+ X

f − X





− f|

− f

(2)

(2z)





−

(2)

(2z)f

(f|

)

(2)

(2z) +

(2)

(2z))

+ X

f −

−

(2)

(2z) −

(4)

(4z)





−

(2)

(2z)f

(f|

)

(2)

(2z) +

(2)

(2z))





− X

+ X



−

(2)

(2z) −

(4)

(4z) +

(f|

)

(2)

(2z) +

(2)

(2z))

= −H

f −

(2)

(2z) +

(f|

)



(2)

(2z))

− f

(2)

(2z) − f

(4)

(4z)



= −H

f −

(2)

(2z) +

(f|

)

(2)

212

thus, we have

−

(2)

(f|

)

(2)

(2z) − H

f −

. ( A. 13)

If we compare the c oeﬃcients of q

and q

2k+1

(for k ≥ 1) of both sides in (A.12), and carry

out som e calculation , we ﬁnd:



− f|

− f

(2)

(2z)







n≥1



−





n≥1



−





n≥1

(2)





n≥1



n≥1

n−1



j=1

n−j

−



n≥1

−



n≥1

(2)

= H



n≥1



n+1



1≤j<n

n−j

− H



−



n≥1

(2)

Thus, for n = 2k and n = 2k + 1 we have t h e following relations

n = 2k : H

2k+1



1≤j<2k

2k−j

− H

−

(2)

= 0;

n = 2k + 1 : H

2k+2



1≤j≤2k

2k−j+1

− H

4k+2

= 0.

Since



1≤j<2k

2k−j

= 2



1≤j<k

2k−j

+ H

and



1≤j≤2k

2k−j+1

= 2



1≤j≤k

2k−j

, in

particular, we obt ai n our ﬁrst two replicable formulas

= H

2k+1



1≤j<k

2k−j



− H

(2)



. (A.14)

4k+2

= H

2k+2



1≤j≤k

2k−j+1

. (A.15)

Develop the other two remaining replication for mulas is a bit more demand i n g. Applying

(A.14) to 2k, k + 1 and 2k + 1 in place of k we ob t a i n

= H

4k+1



1≤j<2k

4k−j



− H

(2)



, (A.16)

4k+4

= H

2k+3



1≤j≤k

2k−j+2



k+1

− H

(2)

k+1



, (A.17)

8k+4

= H

4k+3



1≤j≤2k

4k−j+2



2k+1

− H

(2)

2k+1



; (A.18)

213

and replacing k + 1 for k in (A.15), we also have

4k+6

= H

2k+4



1≤j≤k

2k−j+3

. (A.19)

Now, comparing the coeﬃcients of q

and q

2k+1

(for k ≥ 1) of both sides in (A.13), and

doing some calcula t io n s, we ﬁnd:

−

(2)

(f|

)

(2)

(2z) − H

f −





n≥1







n≥1





n≥1

(2)



−

−H





n≥1



−





n≥1





n≥1

n−1



j=1

2n−2j



n≥1





k≥1

k−1



j=0

4j+2

(2)

k−j

2k+1



k≥1

k−2



j=0

4j+4

(2)

k−j+1



−



n≥1

−



n≥1

= H



n≥1



2n+4



1≤j<n

2n−2j

− H







k≥1

k−1



j=0

4j+2

(2)

k−j

2k+1



k≥1

k−2



j=0

4j+4

(2)

k−j+1



Thus, taki n g n = 2k and n = 2k + 1 above, we have the following relations

n = 2k : H

4k+4



1≤j<2k

4k−2j

k−2



j=0

4j+4

(2)

k−j+1

− H

= 0;

n = 2k + 1 : H

4k+6



1≤j≤2k

4k−2j+2



k≥1

k−1



j=0

4j+2

(2)

k−j

− H

2k+1

− H

8k+4

= 0.

Replacing (A.16) and (A.17) in the expression for 2k above, we have

2k+3



1≤j≤k

2k−j+2



k+1

− H

(2)

k+1





1≤j<2k

4k−2j

k−2



j=0

4j+4

(2)

k−j+1

− H

4k+1

−



1≤j<2k

4k−j

−



− H

(2)



= 0.

214

Thus,

4k+1

= H

2k+3

− H

−



− H

(2)





k+1

− H

(2)

k+1



+ 2



1≤j<k

4k−2j



1≤j≤k

2k−j+2



1≤j<k

4k−4j

(2)

−



1≤j<2k

4k−j

= H

2k+3

− H



+ H

(2)





k+1

− H

(2)

k+1





1≤j≤k

2k−j+2



1≤j<k

4k−4j

(2)





1≤j<k

4k−2j

−



1≤j<2k

4k−j



= H

2k+3

− H



+ H

(2)





k+1

− H

(2)

k+1





1≤j≤k

2k−j+2



1≤j<k

4k−4j

(2)



1≤j<2k

(−1)

4k−j



Similarly, replacing (A.18) and (A.1 9 ) in the expression for 2k + 1 above, we obtain

2k+4



1≤j≤k+1

2k−j+3



1≤j≤2k

4k−2j+2



0≤j<k

4j+2

(2)

k−j

−

−H

2k+1

− H

4k+3

−



1≤j≤2k

4k−j+2

−



2k+1

− H

(2)

2k+1



= 0.

Thus,

4k+3

= H

2k+4

− H

2k+1

−



2k+1

− H

(2)

2k+1





1≤j≤k+1

2k−j+3



0≤j<k

4j+2

(2)

k−j



1≤j≤2k

4k−2j+2

−



1≤j≤2k

4k−j+2

= H

2k+4

− H

2k+1

−



2k+1

− H

(2)

2k+1





1≤j≤k+1

2k−j+3



1≤j≤k

4k−4j+2

(2)





1≤j≤k

4k−2j+2

−



1≤j≤2k

4k−j+2



= H

2k+4

− H

2k+1

−



2k+1

− H

(2)

2k+1





1≤j≤k+1

2k−j+3



1≤j≤k

4k−4j+2

(2)



1≤j≤2k

(−1)

4k−j+2

215

and we have

4k+1

= H

2k+3

− H



+ H

(2)





k+1

− H

(2)

k+1





1≤j≤k

2k−j+2



1≤j<k

4k−4j

(2)



1≤j<2k

(−1)

4k−j

. (A.20)

and

4k+3

= H

2k+4

− H

2k+1

−



2k+1

− H

(2)

2k+1





1≤j≤k+1

2k−j+3



1≤j≤k

4k−4j+2

(2)



1≤j≤2k

(−1)

4k−j+2

. (A.21)

Hence, we have obtained the following four rep l i cat i on formulas (A.14), (A.20), (A.15),

(A.21):

= H

2k+1



1≤j<k

2k−j



− H

(2)



;

4k+1

= H

2k+3

− H



+ H

(2)





k+1

− H

(2)

k+1





1≤j≤k

2k−j+2



1≤j<k

4k−4j

(2)



1≤j<2k

(−1)

4k−j

;

4k+2

= H

2k+2



1≤j≤k

2k−j+1

;

4k+3

= H

2k+4

− H

2k+1

−



2k+1

− H

(2)

2k+1





1≤j≤k+1

2k−j+3



1≤j≤k

4k−4j+2

(2)



1≤j≤2k

(−1)

4k−j+2

To obtain the set of equations (7.19)-(7.22), it is just a change of notation. Recall that

we write c

(n) to be the coeﬃcient of q

in the McKay-Thompson series T

(z)

′

of g ∈ M,

that is T

(z)

′

= q

−1



n≥1

(n)q

, with q = e

2πiz

. Also, r e cal l Conway and Norton

already proved in [38] that the s-th replication associated to T

(z)

′

is exactly T

(z)

′

, the

McKay-Thompson series associated to g

. Hence, taking f = T

′

, we have the following

dictionary

= c

(n) and H

(2)

= c

(n),

from wh er e the replication formulae (7.19)-(7.22) directly appear.

Also, note that when comparin g coeﬃcients of q

and q

2k+1

in (A.13), we have obtained

in add i t i on

= H

−

(2)

that i s, equation (7.23), using notation c

(n).

216

Index

adjoining the derivation, 36

adjoint map, 28, 51

adjoint representation, 51

algebra

aﬃne, 46

extended, 46

extended twisted, 48

twisted, 47

untwisted, 46

associative, 27

commutative, 27

graded, 38

nonassociative, 27

quotient, 34

Atlas of ﬁnite groups, 4, 5

automorphism, 28

of vertex algebras, 93

B, see Griess algebra

Baby Monster, 3, 5, 6

base

of a root system, 63

Bernoulli numbers, 140

bihomomorphism, 38

Bimonster, 7

Borcherds

algebra, 165

character formula, 169

denominator identity, 170

identity, 160

, see Golay code

Cartan matrix , 88

generalized, 89

of aﬃne type, 90

of ﬁnite type, 90

of indeﬁn it e type, 90

symmetrisable, 90

Cartan sub algebr a, 57

center

of a Lie algebra, 35

character, 66, 88

table, 5

Chevalley basis, 100

chiral algebra, 195

class func ti on, 66

co-root, 59

real, 67

code, 105

of type I, 105

of type II, 105

self-dual, 106

code loops, 125

commensurable, 15

commutator ideal, 35

compact

matrix Lie group, 25

real form, 56

complete reducibility property, 53

completely reducible, 39, 53

complexiﬁcation

of a Lie algebra, 32

of a vector space, 32

conformal

ﬁeld th eor y, 195

vector, 84

vertex algebra, 85

congruence subgroup, 153

connected

matrix Lie group, 25

contravariant bilinear form, 165

convergence

in matrix groups, 22

Conway group

, 110

Conway sporadic group

, 3, 5

, 3–5

correspondence, 149

Coxeter graph, 7

associated group to, 7

cusp, 10

cusp form, 130

Dedekind’s η f un ct ion, 10

degree operator, 36

denominator identity

twisted, 182

derivation, 28

direct produ ct

217

of Lie algebras, 35

direct s u m

of modules, 39

discriminant, 134

dual

code, 105

lattice, 94

root system, 63

Dynkin diagram, 98

Eisenstein series, 10, 133, 142

endomorphism, 28

equivalence

of Dynk in diagrams, 98

of extens ions , 35

of modules, 34

of repr es e ntations, 50

exponencial matr ix , 29

extension

central, 35

of Lie algebras, 35

trivial, 35

Faber polynomial, 207

ﬁnite gr oups

classiﬁcation of, 2, 4

Fischer sporadic group

, 3, 5

Fi’

, 3, 5

Fricke element, 186

fundamental functions, 10

genus

of a group, 15

Golay cod e, 4, 106

pqr

, see Coxeter graph

graded

algebra, 38

character, 179

dimension, 179

module, 39, 123

subspace, 36

vector space, 36

graded component, 36

graded dimen si on, 13

grading

of a tensor product, 38

grading pr es er v in g, 36

Griess algeb r a, 4, 116

Griess module, 114

group

alternating, 2

Chevalley, 2

classiﬁcation project , 2

Coxeter, 7

cyclic, 2

Monster, see Monster

of Lie type, 2

of Monster - typ e , 125

simple, 2

sporadic, 2, 3

group algebr a, 41

Haar measure , 54

Hall divi sor , 154

Hall-Janko sporadic group HJ, 3

Hamming code, 106

Happy family, 4

Haputmodul

of a group, 11

Harada-Norton spor adic group HN, 3, 5

Hauptmodul, 10

normalized, 11, 15

Hecke operator, 150

Heisenberg group, 24

Held sporadic group He, 3

higher, 65

highest weight, 65

Higman-Sims spor adic group HS, 3

holomorphic at inﬁnity, 130

homogeneous

space of degree α, 36

linear map, 36

subspace, 36

homomorphism, 28

of Lie algebras, 28

of modules, 34

ideal, 34

central, 35

left, 34

right, 34

indecomposable

module, 34

induced

module, 40, 41, 43

integral

dominant integral, 64

integral element, 63, 67

intertwining map, 50

invariant

218

bilinear map, 45

subspace, 34, 49

involution, 47

irreducible, 49

module, 34

isometry, 94

isomorphism, 28

of lattices, 94

of modules, 34

j function, 1, 139

Janko sp or adic group

, 3, 5, 6

, 3–5

, 3, 4

Jordan-H¨older theorem, 2

Kac-Moody algebra, 89

, see Leech lattice

lattice, 93

even, 94

integral, 94

non degener ate, 94

positive deﬁnite, 94

self-dual, 94

Leech lattice, 4–6, 108

shorter, 191

Lie algebra, 28

commutative, 28

of a Lie group, 29

quotient, 34

reductive, 55

semisimple, 55

Lie group, 26

linear c ode, 105

linear gr oup

general, 22

special, 22

Lorentzian space R

25,1

, 4

Lyons spor adic group Ly, 3, 4

M, see Monster

Mathieu group

, 3, 5

Mathieu sporadi c group

, 3

, 3, 5

, 4

Matieu group M

, 107

matrix Lie group, 22

McKay-Thompson series, 14

McLaughlin sp oradi c group McL, 3

meromorphic at inﬁnity, 130

mirror symmetry, 194

Mirror-Moonshine conjecture, 195

modular

equation, 160

form, 10, 130

function, 10, 130

lattice function, 132

weakly, 130

modular group, 1, 8, 127

fundamental domain, 128

module, 33, 41

Monster, 1, 2, 5, 6

irreducible character degrees, 4

Lie algebras , 171

representation of, 7

size of, 5

Monster vertex algebr a, 114

Moonshine

generalized, 185

Moonshine conjec tu r e, 1

Moonshine module, 1, 113

moonshine-type, 154

multiplicity, 176

of weights, 64

Niemeier lattice, 111

of type A

, 111

no-ghost t he ore m, 171

normal ord er e d product, 84

Norton se r ie s , 186

O’Nahn s poradic group O’N, 3, 4

octad, 106

orbifold, 185

order

of f at p, 135

orthogonal gr oup , 23

complex speci al, 23

special, 23

orthonormal group

complex, 23

pariah, the, 4

product

in an algebra, 27

219

quilt, 188

quotient

module, 39

Ramanujan’s τ function, 143

rank

of a Lie algebra, 58

replicable function, 160

replication formulae, 158, 159, 161

replication power, 209

representation

adjoint, 34

basic, 91

complex, 49

faithful, 49

left re gular , 41

modular, 192

of a group, 41

of Lie algebras, 33

of modules, 33

real, 49

trivial, 41

root, 58, 168

imaginary, 90, 168

negative, 63

positive, 63, 168

real, 66, 90, 168

simple, 63, 168

system, 62

root space, 58

root vector, 58

Rudvalis sporadic group Ru, 3, 4

semidirect pro d uc t

of Lie algebras, 35

semisimple

module, 39

sextet, 106

shorter Moonshi ne module, 191

simple

module, 34

simple Li e algebra, 34

simply c onne ct ed

matrix Lie group, 26

skew-symmetry, 28

(Z), s ee modular group

sporadic groups

list of all, 3

standard representation

of a Lie group, 50

of Lie algebras, 51

string, 195

subalgebra, 28

submodule, 34

Suzuki sporadi c group Suz, 3

symmetric algebra, 42

symmetric p owers, 43

symplectic group

compact, 24

complex, 24

real, 24

tensor algebra, 42

tensor map, 37, 38

tensor multiplication, 37

tensor produ ct, 41

module, 39

of vector spaces, 37

theta f un ct ion, 96, 146

Third gen er ati on, 4

Thompson

order for mula, 5

sporadic group Th, 3, 5

totally singular, 106

transform, 149

unimodular

lattice, 94

unimodular Lie group, 54

unitary matrix, 23

universal Borcherds algebra, 166

universal enveloping algebra, 43

♮

, see Moonshine module

vertex algebra, 81

vertex operator, 82

vertex operator algebra, 1, 85

Virasoro algebr a, 86

Weierstrass ℘ function, 134

weight, 64, 76

fundamental, 78

of a code, 105

real, 67

space, 64

vector, 64, 76

weight distribution, 106

Weyl

character formula, 67

group, 168

Weyl group, 60

220

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