Table Of ContentAutomorphic Forms and Lie Superalgebras
Algebra and Applications
Volume 5
Managing Editor:
Alain Verschoren
University of Antwerp, Belgium
Series Editors:
Eric Friedlander
Northwestern University, U.S.A.
John Greenlees
Sheffield University, U.K.
Gerhard Hiss
Aachen University, Germany
Ieke Moerdijk
Utrecht University, The Netherlands
Idun Reiten
Norwegian University of Science and Technology, Norway
Christoph Schweigert
Hamburg University, Germany
Mina Teicher
Bar-llan University, Israel
Algebra and Applications aims to publish well-written and carefully refereed
monographs with up-to-date expositions of research in all fields of algebra,
including its classical impact on commutative and noncommutative algebraic and
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related domains, such as number theory, homotopy and (co)homology theory
through to discrete mathematics and mathematical physics.
Particular emphasis will be put on state-of-the-art topics such as rings of differential
operators, Lie algebras and super-algebras, group rings and algebras, Kac-Moody
theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well
as their applications within mathematics and beyond. Books dedicated to
computational aspects of these topics will also be welcome.
Automorphic Forms
and Lie Superalgebras
by
Urmie Ray
Université de Reims, Reims, France
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-5009-7 (HB)
ISBN-13 978-1-4020-5009-1 (HB)
ISBN-10 1-4020-5010-0 (e-book)
ISBN-13 978-1-4020-5010-7 (e-book)
Published by Springer,
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© 2006 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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To my Mother
Contents
Preface ix
1 Introduction 1
1.1 The Moonshine Theorem . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 A Brief History . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Borcherds-Kac-Moody Lie Superalgebras . . . . . . . . . . . . . . 5
1.3 Vector Valued Modular Forms . . . . . . . . . . . . . . . . . . . . 7
1.4 Borcherds-Kac-Moody Lie Algebras
and Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Γ-graded Vertex Algebras . . . . . . . . . . . . . . . . . . . . . . 9
1.6 A Construction of a Class of
Borcherds-Kac-Moody Lie (super)algebras . . . . . . . . . . . . . 9
2 Borcherds-Kac-Moody Lie Superalgebras 13
2.1 Definitions and Elementary Properties . . . . . . . . . . . . . . . 13
2.2 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 The Root System . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Uniqueness of the Generalized Cartan
Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 A Characterization of BKM Superalgebras . . . . . . . . . . . . . 66
2.6 Character and Denominator Formulas . . . . . . . . . . . . . . . 72
3 Singular Theta Transforms of Vector Valued Modular Forms 93
3.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2 Ordinary Modular Functions . . . . . . . . . . . . . . . . . . . . 95
3.3 Vector Valued Modular Functions . . . . . . . . . . . . . . . . . . 102
3.4 The Singular Theta Correspondence . . . . . . . . . . . . . . . . 113
4 Γ-Graded Vertex Algebras 129
4.1 The Structure of Γ-graded Vertex Algebras . . . . . . . . . . . . 129
4.2 Γ-Graded Lattice Vertex Algebras . . . . . . . . . . . . . . . . . 149
4.3 From Lattice Vertex Algebras to Lie Algebras . . . . . . . . . . . 167
vii
viii Contents
5 Lorentzian BKM Algebras 177
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.2 Automorphic forms on Grassmannians . . . . . . . . . . . . . . . 179
5.3 Vector Valued Modular forms and LBKM Algebras . . . . . . . . 186
5.4 An Upper Bound for the Rank of the Root Lattices
of LBKM Algebras? . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.5 A Construction of LBKM Algebras
from Lattice Vertex Algebras . . . . . . . . . . . . . . . . . . . . 213
A Orientations and Isometry Groups 239
B Manifolds 241
B.1 Some Elementary Topology . . . . . . . . . . . . . . . . . . . . . 241
B.2 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
B.3 Fibre Bundles and Covering Spaces . . . . . . . . . . . . . . . . . 247
C Some Complex Analysis 251
C.1 Measures and Lebesgue Integrals . . . . . . . . . . . . . . . . . . 251
C.2 Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 253
C.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
C.4 Some Special Functions . . . . . . . . . . . . . . . . . . . . . . . 255
D Fourier Series and Transforms 263
Index 283
Preface
I would like to thank Profs. M. Castellet, G. Cornelissen, and E. Friedlander,
for their encouragements at the start of this project. I would like to take this
occasion to express my gratitude to Prof. R. Borcherds for many stimulating
discussions and for his advice and support throughout the years. I am also
gratefultohimandtoProfs. V.Kac,J.LepowskyandC.Schweigertforreading
partofthemanuscriptandfortheirvaluablecomments. Themajorpartofthis
book was written while I was on leave at the Centre de Recerca Matema`tica,
in Barcelona, Spain. I express my thanks to the staff of the institute for their
hospitality and to the Ministerio de Educacio´n of the Spanish government for
providing financial support during this period.
Most of all, I am indebted to Dr. C.-M. Patris whose unwavering support
and trust made it possible for me to finish this book.
ix
Chapter 1
Introduction
In this book, we give an exposition of the theory of Borcherds-Kac-Moody Lie
algebras and of the ongoing classification and explicit construction project of a
subclass of these infinite dimensional Lie algebras. We try to keep the material
as elementary as possible. More precisely, our aim is to present some of the
theory developed by Borcherds to graduate students and mathematicians from
other fields. Some familiarity with complex finite dimensional semisimple Lie
algebras,grouprepresentationtheory,topology,complexanalysis,Fourierseries
and transforms, smooth manifolds, modular forms and the geometry of the
upperhalfplanecanonlybehelpful. However,eitherintheappendicesorwithin
specific chapters, we give the definitions and results from basic mathematics
needed to understand the material presented in the book. We only omit proofs
of properties well covered in standard undergraduate and graduate textbooks.
There are several excellent reference books on the above subjects and we
will not attempt to list them here. However for the purpose of understanding
theclassificationandconstructionofBorcherds-Kac-Moody(super)algebrasthe
following are particularly useful. Serre’s approach to the theory of finite di-
mensional semisimple Lie algebras in [Ser1] is conducive to the construction of
Borcherds-Kac-Moody Lie algebras as it emphasizes the presentation of finite
dimensional semi-simple Lie algebras via generators and relations. For a first
approach to automorphic forms and the geometry of the upper half plane, the
book by Shimura may be a place to start at [Shi].
We also do not replicate the proofs of properties of Kac-Moody Lie algebras
that are treated in depth in the now classical reference book by Kac [Kac14].
Borcherds-Kac-MoodyLiealgebrasaregeneralizationsofsymmetrizable(see
Remarks 2.1.10) Kac-Moody Lie algebras, themselves generalizations of finite
dimensional semi-simple Lie algebras. That this further level of generality
is needed was first shown in the proof of the moonshine theorem given by
Borcherds[Borc7],wherethetheoryofBorcherds-Kac-MoodyLiealgebrasplays
a central role. So let us briefly explain what this theorem is about.
1
2 1 Introduction
1.1 The Moonshine Theorem
The remarkable Moonshine Theorem, conjectured by Conway and Norton and
onepartofwhichwasprovedbyFrenkel,LepowskyandMeurman,andanother
by Borcherds, connects two areas apparently far apart: on the one hand, the
Monster simple group and on the other modular forms. Any connection found
between an object which has as yet played a limited abstract role and a more
fundamental concept is always very fascinating. Also it is not surprising that
ideasonwhichtheproofofsucharesultarebasedwouldgiverisetomany new
questions,thusopeningupdifferentresearchdirectionsandfindingapplications
in a wide variety of areas.
1.1.1 A Brief History
The famous Feit-Thompson Odd order Theorem [FeitT] implying that the only
finitesimplegroupsofoddorderarethecyclicgroupsofprimeorder,sparkedoff
much interest in classifying the finite simple groups in the sixties and seventies.
The classification of the building blocks of finite symmetries was completed in
the eighties [Gor]: there are 17 infinite families of finite simple groups and 26
which do not belong to any families and are thus called sporadic [Bur]. The
MonsterM [Gri]isthelargestsporadicsimplegroup. Ithasabout1054elements
or more precisely
246.320.59.76.112.133.17.19.23.29.31.41.47.59.71
elements. Evidence for the existence of the Monster were first found by B.
Fischer and R.L. Griess in 1973. Based on Conway and Norton’s conjecture
that the dimension of one of its representations is 196883 Fischer, Livingstone
and Thorne computed the full character table of M [FisLT]. We remind the
reader that a group representation is a group homomorphism from the group
to the isomorphism group of a vector space. Then, McKay noticed that the
dimensionsofitstwosmallestrepresentations, 1and196883, arecloselyrelated
tothefirsttwocoefficientsoftheFourierexpansionsofthenormalizedmodular
invariant
J =j−744.
Since j(τ +1)=j(τ) for τ in the upper half plane H={x+iy|y >0}, we can
write j as a function of q = e2πiτ. Moreover j is holomorphic on H. So, as a
function of q, J has a Laurent series in the punctured disc of radius 1 centered
at 0:
(cid:1)∞
J(q)=j(q)−744=q−1+196884q+...= c(n)qn
n=−1
and c(−1) = 1, c(1) = 196883+1. This is the Fourier series of J. Further
details can be found in [Ser2].
At the time it was thought unlikely that there would be any connection
between modular forms and the Monster group. Shortly after, McKay and
