The Monster and Lie Algebras: Proceedings of a Special Research Quarter at the Ohio State University, May 1996 9783110801897, 9783110161847

Table of contents :
Preface
Part I The Monster
Vertex operators in algebraic topology
The radical of a vertex operator algebra
Associative subalgebras of the Griess algebra and related topics
A vertex operator algebra related to E8 with automorphism group O+(10,2)
Modular forms associated with the Monster module
Quilts, the 3-string braid group, and braid actions on finite groups: an introduction
The moonshine VOA and a tensor product of Ising models
Netting the Monster
Monster roots
Part II Lie Algebras
On graded Lie algebras of characteristic three with classical reductive null component
Auslander-Reiten theory for restricted Lie algebras
Chief factors and the principal block of a restricted Lie algebra
On Drinfeld realization of quantum affine algebras
Free Lie superalgebras and the generalized Witt formula
A generalization of the Jordan approach to symmetric Riemannian spaces
Representation theory of Lie algebras of Cartan type

Citation preview

Ohio State University Mathematical Research Institute Publications 7 Editors: Gregory R. Baker, Walter D. Neumann, Karl Rubin

Ohio State University Mathematical Research Institute Publications 1 2 3 4 5 6

Topology '90, B. Apanasov, W. D. Neumann, A. W. Reid, L. Siebenmann (Eds.) The Arithmetic of Function Fields, D. Goss, D. R. Hayes, Μ. I. Rosen (Eds.) Geometric Group Theory, R. Charney, M. Davis, M. Shapiro (Eds.) Groups, Difference Sets, and the Monster, Κ. T. Arasu, J. F. Dillon, K. Harada, S. Sehgal, R. Solomon (Eds.) Convergence in Ergodic Theory and Probability, V. Bergelson, R March, J. Rosenblatt (Eds.) Representation Theory of Finite Groups, R. Solomon (Ed.)

The Monster and Lie Algebras Proceedings of a Special Research Quarter at The Ohio State University, May 1996

Editors

J. Ferrar K. Harada

w DE

G_ Walter de Gruyter · Berlin · New York 1998

Editors JOSEPH FERRAR, KOICHIRO HARADA

Department of Mathematics, The Ohio State University 231 West 18th Avenue, Columbus, OH 43210, USA Series Editors: Gregory R. Baker Department of Mathematics, The Ohio State University, Columbus, Ohio 43210-1174, USA Karl Rubin Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA Walter D. Neumann Department of Mathematics, The University of Melbourne, Parkville, VIC 3052, Australia 1991 Mathematics Subject Classification: 20-06, 17-06; 17Bxx ©

Printed on acid-free p a p e r which falls within the guidelines of the A N S I to ensure p e r m a n e n c e a n d durability.

Library of Congress - Cataloging-in-Publication

Data

The Monster and Lie algebras : proceedings of a special research quarter held at the Ohio State University, Spring 1996 / editors, J. Ferrar, K. Harada. p. cm. - (Ohio State University Mathematical Research Institute publications ; 7) ISBN 3-11-016184-2 (alk. paper) 1. Lie algebras - Congresses. 2. Vertex operator algebras - Congresses. 3. Group theory — Congresses. 4. Mathematical physics - Congresses. I. Ferrar, J. (Joseph) II. Harada, Koichiro, 1941 - . III. Series QC20.7.L54M66 1998 512'.55-dc21 98-29397 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication

Data

The Monster and Lie algebras : proceedings of a special research quarter at The Ohio State University, May 1996 / ed. J. Ferrar ; K. Harada. - Berlin ; New York : de Gruyter, 1998 (Ohio State University Mathematical Research Institute publications ; 7) ISBN 3-11-016184-2

© Copyright 1998 by Walter de Gruyter GmbH & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typeset using the authors' T E X files: I. Zimmermann, Freiburg. Printing: WB-Druck GmbH & Co., Rieden/Allgäu. Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

Preface

These Proceedings are an outgrowth of a conference culminating a special research quarter held at the Ohio State University in Spring 1996 and supported by the O.S.U. Mathematical Research Institute and the National Science Foundation. The focus of the conference was groups, Lie algebras and the Monster, and considerable emphasis was placed on presenting the various aspects of group theory and Lie algebra theory in a modern perspective. The Monster served nobly as a central theme with deep roots in both Lie algebra and group theory. The papers in this volume represent only a portion of the material presented at the conference but they give a fascinating picture, in both expository presentations and state-of-the-art research reports, of the breadth and vitality of current activity in these areas and of the extraordinary interconnections that have repeatedly surfaced in their study. It is a pleasure to acknowledge the efforts of a number of people who helped make the special quarter and conference successful. The Ohio State University Mathematics Department staff—particularly Marilyn Radcliffe—provided much needed organizational support. Terry England played an important role in the preparation of manuscripts for publication. Professor Tom Gregory shared much of the responsibility for planning the special quarter and the conference, and for hosting long-term visitors. The Monster. Three years after the first, the second Columbus Monster conference was held in May, 1996. In the preface of the proceedings of the first Columbus Monster Conference (de Gruyter, Berlin-New York, 1996), I wrote: "The moonshine has not yet turned into the sunshine, far from it actually. Our progress, however, is slow and steady. The year 1993, when the conference was held, is the 20th year since the Monster first appeared in the world. The Monster is now an adult at least physically if not mentally.' The Monster now is 24 years old and the moonshine is still a moonshine. The Monster, however, has not exhausted its supply of surprises for us. Given below is a brief description of each individual work submitted for these proceedings. A. Baker contributed a paper which is to give algebraic topologists an introduction to some of the algebraic ideas associated with vertex operator algebras and to show to algebraists that the vertex operator algebras have topological significance, especially in Conformal Field Theory. Vertex operator algebras have appeared in connection with elliptic genera and elliptic cohomology also. Dong-Li-Mason-Montague investigates the radical of a vertex operator algebra V. If ν is an element of the homogeneous component V* of V, then the endomorphism vn maps Vm to Vm+k-n-\• In particular, the zero mode o(v) = v^-i acts on V„ for each η € Ζ. The radical J( V) of the vertex operator algebra V is defined by J(V) = {υ € V I o(v) = 0}.

vi

Preface

They show that under a certain condition for V, J ( V ) can be determined. Dong-Li-Mason-Norton constructs many interesting (maximal) associative subalgebras in the weight 2 subspace V2 of the moonshine module Vc. The key idea of Dong-Li-Mason-Norton is to relate this question to the theory of root systems and the Niemeier lattices. For example, the L(^,0)® 2 4 corresponds to the lattice A2{4. R. L. Griess constructs a vertex operator algebra with automorphism group Ο + (10, 2), one of the 3-transposition groups. T. Hsu deals with quilts (developed by Norton, Parker, Conway, and Hsu). Some basic definitions and results are first given. He shows how quilts may be used to study an action of B^, the 3-string braid group, on pairs of elements of a finite group. The theory of quilts may be relevant to Norton's generalized moonshine conjectures though no strong evidence is yet to been seen. Some examples are worked out. M. Miyamoto has defined automorphisms for a vertex operator algebra which contains an element ν whose components vn of the vertex operator Y(v, z) generate a Virasoro algebra of central charge \ . If such an element exists, then using the fusion rule of the rational vertex operator algebra 0), one can construct an involutive automorphism on V. Miyamoto's initial idea to produce involutive automorphisms appears to be able to give rise to automophisms of order larger than 2. S. Norton has been involved in the Monster/moonshine since its birth some 24 year ago. His name is associated with such notions as the generalized moonshine conjecture, quilts, and of course, the epoch-making paper of Conway-Norton. In the paper submitted for these proceedings, Norton discusses his new game called net, which is a finite geometry associated with the triple (a, b,c) where a, b,c are three involutions of a group G. The subgroup ( a , b , c ) is called the net group. C. Simons's paper deals the Y-diagram whose group of automorphisms is isomorphic to the Bimonster, the wreathed product of Μ by a cyclic group of order 2, the fact first being conjectured and then proved, by a combined effort of J. H. Conway, A. D. Pritchard, S. P. Norton, A. A. Ivonov. Let τ be the wreathing involution of the Bimonster. r is called the reflection element of the Bimonster. The generators of the Y-diagram correspond to reflections called fundamental Monster roots. Simons shows that the naming relationship will provide an easy way to compute with the Monster. The conference was stimulating for all participants. The speakers and their topics were the following (in the order of presentation): John McKay, Concordia University (Canada), Some indiscrete thoughts about the Monster, Alex Ryba, University of Minnesota, Modular moonshine', Ching Hung Lam, Ohio State University, Vertex operator algebras with V\ = 0; Robert L. Griess, Jr., University of Michigan, Automorphisms

ofVOAs;

Simon Norton, University of Cambridge (U.K.), Footballs and the Monster; Geoffrey Mason, University of California at Santa Cruz, (Colloquium talk) Modular functions of genus zero and the generalized moonshine conjectures',

Preface

Elizabeth Jurisich, University of Chicago, On integrable modules for Kac-Moody algebras;

vii

generalized

Masahiko Miyamoto, University of Tsukuba (Japan), Tensor products of Ising models and the moonshine module; Steven Smith, University of Illinois at Chicago, Finite-geometry structures associated with the Monster; Gerald Hoehn, University of California at Santa Cruz, Self-dual codes, lattices and vertex operator algebras — a fourfold analogy; Chongying Dong, University of California at Santa Cruz, Compact groups of vertex operator algebras;

automorphism

Haisheng Li, University of California at Santa Cruz, Certain associative similar to Uislj) and Zhu's algebra A(V/J ;

algebras

Chris Cummins, Concordia University (Canada), Modular equations and moonshine functions; Michael P. Tuite, University College, Galway (Ireland) and Dublin Institute for Advanced Studies, Dublin (Ireland), Moonshine for Siegel forms?; Paul S. Montague, University of California at Santa Cruz, On the self-duality of the Monster module and other lattice orbifold theories; Timothy Hsu, Princeton University, Quilts and braid actions on elements of finite groups: an overview; Chris Simons, Princeton University, M666 and Monster roots. Koichiro Harada

Lie Algebras. 1996 marked the 25th anniversary of a Conference on Lie Algebras held at the Ohio State University and featuring presentations by Nathan Jacobson and A. Adrian Albert. At the time of that first conference, Kac-Moody algebras were in their infancy, Lie superalgebras were yet to appear on the scene, the classification of simple modular Lie algebras was but a wishful thought, and quantum groups were not yet conceived. By the date of this most recent conference, much of this had changed, and with it the thrust of research efforts directed towards the study of Lie algebras and related topics. The goal of the conference organizers was to represent as broad a view as possible of the directions in which recent advances have pushed the study of Lie algebras. Presentations covered topics from geometry, representation theory, modular Lie algebras of low characteristic, (extended) affine Lie algebras, quantum groups, groups associated with Lie algebras, and applications of Hopf algebras in the study of Lie algebras. The papers contributed to these proceedings are a representative cross section of the invited presentations. They are well-suited for a reader interested in surveying current research trends associated with Lie algebras.

viii

Preface

A complete list of speakers, and their topics, for this portion of the conference follows: M. Kuznetsov, Nizhny Novgorod State University (Russia), Simple modular graded Lie algebras', G. Benkart, University of Wisconsin, Madison, Towards a representation theory for Lie algebras graded by finite root systems; S. J. Kang, Seoul National University (Korea), Denominator identity for graded Lie algebras and Lie superalgebras; K. Misra, N. Carolina State University, Demazure modules for the affine Lie algebra si(2); R. Farnsteiner, University of Wisconsin, Milwaukee, Auslander-Reiten restricted Lie algebras',

theory for

J. Feldvoss, Hamburg University (Germany), Burnside 's Theorem for restricted Lie algebras', D. Nakano, Utah State University, Support varietiesfor induced modules over quantum groups; I. Kantor, Lund University (Sweden), A generalization of a Jordan approach to Riemannian symmetric spaces; J. Faulkner, University of Virginia, Elementary groups for Kantor pairs; R. Griess, University of Michigan, Recent progress in finite subgroups of Lie groups; B. Allison, University of Alberta (Canada), Extended affine Lie algebras; I. Kryliouk, Gainesville, Florida, The automorphism groups of some quasi-simple Lie algebras; R. Block, University of California, Riverside, Α Hopf algebra generalization of the elimination theory for free Lie algebras; N. Jing, N. Carolina State University, On twisted quantum affine algebras. Joe Ferrar

Table of Contents

Preface

ν

Parti The Monster Andrew Baker Vertex operators in algebraic topology

3

Chongying Dong, Haisheng Li, Geoffrey Mason, and Paul S. Montague The radical of a vertex operator algebra

17

Chongying Dong, Haisheng Li, Geoffrey Mason, and Simon P. Norton Associative subalgebras of the Griess algebra and related topics

27

Robert L. Griess, Jr. A vertex operator algebra related to Eg with automorphism group 0 + ( 10,2)

43

Koichiro Harada and Mong Lung Lang Modular forms associated with the Monster module

59

Tim Hsu Quilts, the 3-string braid group, and braid actions on finite groups: an introduction

85

Masahiko Miyamoto The moonshine VOA and a tensor product of Ising models

99

Simon P. Norton Netting the Monster

Ill

Christopher S. Simons Monster roots

127

Part II Lie Algebras Georgia Benkart, Thomas Gregory, and Michael I. Kuznetsov On graded Lie algebras of characteristic three with classical reductive null component

149

χ

Table of Contents

Rolf Farnsteiner Auslander-Reiten theory for restricted Lie algebras

165

Jörg Feldvoss Chief factors and the principal block of a restricted Lie algebra

187

Naihuan Jing On Drinfeld realization of quantum affine algebras

195

Seok-Jin Kang Free Lie superalgebras and the generalized Witt formula

207

Issai L. Kantor A generalization of the Jordan approach to symmetric Riemannian spaces

221

Daniel K. Nakano Representation theory of Lie algebras of Cartan type

235

PART I THE MONSTER

Vertex operators in algebraic topology Andrew

Baker

Introduction This paper is intended for two rather different audiences. First we aim to provide algebraic topologists with a timely introduction to some of the algebraic ideas associated with vertex operator algebras. Second we try to demonstrate to algebraists that many of the constructions involved in some of the most familiar vertex operator algebras have topological (and indeed geometric) significance. We hope that both of these mathematical groups will benefit from recognition of their links in this area. Rather than simply attempting to survey the area, we have reworked some aspects to emphasise integrality and other algebraic features that are less well documented in the literature on vertex operator algebras, but probably well understood by experts. The notion of a vertex operator algebra is due to R. Borcherds and arose in the algebraicization of structures first uncovered in the context of Conformal Field Theory and representations of infinite dimensional Lie algebras and groups. A spectacular example is provide by the Monster vertex operator algebra, V*, whose automorphism group is the Monster simple group M. As well as the book of Frenkel, Lepowsky and Meurman [5], the paper of Dong [2] and the memoir of Frenkel, Huang and Lepowsky [4] provide algebraic details on vertex operator algebras, and we take these as basic references. The work of [6] already gives a hint that there is an 'integral' structure underlying some of the algebraic aspects of Conformal Field Theory. In this paper we will show that there are integral (at least after inverting 2) structures within some of the most basic examples of vertex operator algebras associated to positive definite even lattices. We will also interpret such algebras in terms of the (co)homology of spaces related to the classifying space of ^-theory. In future work we will further clarify the topological connections by explaining their origins in the geometry of certain free loop spaces as described in the work of Pressley and Segal [9], [8]. Although our topological interpretation of vertex operator algebras involves homology, it could just as easily (and perhaps more naturally) be given in terms of cohomology. We could even describe such structures in generalized (co)homology theories, particularly complex oriented theories. There is some evidence that vertex operators may usefully be viewed as giving rise to families of unstable operations in such theories, perhaps leading to algebraic generalizations of vertex operator algebras appropriate to the study of some important examples, and we intend to consider these issues in future work. For the benefit of topologists we note that vertex operator algebras have appeared in work of H. Tamanoi and others in connection with elliptic genera and elliptic cohomology as well the study of loop spaces and particularly loop groups. Indeed it is possible that

4

A. Baker

a geometric model of elliptic cohomology will involve vertex operator algebras and their modules. My understanding of the material in this paper owes much to the encouragement and advice of Jack Morava and Hirotaka Tamanoi, as well as the many preprints supplied by Geoff Mason and Chongying Dong. I also wish to acknowledge financial support from the EU, University of Glasgow, IHES, NSF, Ohio State University and Osaka Prefecture. Finally I would like to thank Koichiro Harada for organising the timely Ohio meeting for the Friends of the Monster.

§1 Vertex operator algebras and their modules Let Ik be a field of characteristic 0. Let V — V, denote a Z-graded vector space over k; following [5], we denote the η th grading by V( n) and for ν e V( n) we refer to η as the weight of ν and write wt ν = η. Whenever we refer to elements of V, we always assume that they are homogeneous. Suppose that there is a k-linear map Y( > ζ): V — • Endk(V)[[z, z - 1 ] ] , where for any abelian group Μ, M[[z,z~1]]

= {j2mnZn:mn€M nez

J.

We write

γ (υ,ζ) = Συ»ζ~η~1> neZ

where vn e Endk(V), vnu = vn(u) and

ηεΖ The pair (V, Y) gives rise to a vertex operator algebra if the following axioms are satisfied VOA-1 VOA-2 VOA-3 VOA-4 VOA-5

For each η e Ζ, dim^ V( n) < οο. For η 0, dim k V(„) = 0. Given elements u, ν e V, unv = 0 for 0 n. There are two distinguished elements 1, ω e V and a rational number rank V. We set ω η + ι = L n . For any ν e V, we have the identities Y ( l , z) = Id v ; Y ( w , z ) l e V[[z]]; Υ(υ, 0)1 = lim Υ(υ, ζ) 1 = υ. ζ—

Vertex operators in algebraic topology

VOA-6

The following identity amongst operator valued Laurent series in the variables zo. z\, Z2 holds: Ζο"1'

Y(M

'

Ζι)Υ(υ

'Zl)

(^f") γ(υ'Z2)Y(M'Zl)

-

= ζϊιδ

VOA-7

γ

( γ ( " . ζο)υ, Z2),

where the expansion of the Dirac function δ will be discussed below. The elements Ln (as operators on V) satisfy [Lm, L„] = (m - n)Lm+n

VOA-8

5

Η

(m3 — m) — (rank V) neZ

(1.1)

6

A. Baker

and for three variables zo, ζ ι, zi,

" E S - ' ^ k r t ·

(1-2)

In other words, we expand in terms of the second variable in the numerator of the argument. From [2] and [5], we also record the definition of a module (3Vt, Ym) over a vertex operator algebra (V, Y, 1, ω, rank V). This consists of a Q-graded k-module Μ = Μ . together with a k-linear map Y M : V — * E n d ( M )[[z,z~{]]; V I • Ym(W.Z) = neZ

satisfying the following conditions. VOM-1 VOM-2 VOM-3 VOM-4

For each η e Q, dim k M(„) < oo. For η « 0, dimk M ( n ) = 0. Given elements u , v e V , u^v — 0 for 0 We have the identity

n.

Y M ( l , z ) = Id M . VOM-5 The following identity amongst Laurent series in z0,zi,z2 holds: ^

(Zl

Z2

ZQ

)

Z2) -

Zq" 1 ^ ^ Z 2 _ Z q Z ' )

= zj1^

Z Z2

Z2)Ym(",

° ) Y3vt(Yv(w, zo)v, Zi).

VOM-6 The elements Ln = ωη+ι (as operators on M ) satisfy [Lm > ] — (m — n)Lm+n Η VOM-7

—

zi)

(rank V)ο bnT" is a grouplike element of H*(BU)[[T]]. There are also the primitive elements pn e H2n(BU), satisfying the Newton recurrence relation P 1 =

Pn

'1· = bipn-l

~ b2pn-2

+b3Pn-3

K-lf

2

bn-\p\+(-\)n

,

nbn.

(2.1)

When k is a Q-algebra, this is equivalent to the generating function identity

where ln(l + Ζ) = Σ η > ι ( — l ) n ~ l Z n / n is the formal logarithmic series. The primitive submodule of Ηιη (BU) is generated by pn, but these elements are not algebra generators of H*(BU) over a general ring k, in particular over integers Z. They do however generate over the rationals Q, which accounts for the fact that in [5] they are used in giving explicit formulae for vertex operators.

8

A. Baker

Dually, in H*(BU) we have the universal Chern classes cn G Hln(BU), which are also polynomials generators and generate a binomial coalgebra by the Cartan formula. The duality is given by the formula ι

i f • • • b r k k = bi,

0

else.

The primitive submodule in H2n (BU) has generator sn satisfying the recurrence relation S1 = C\, Sn = Ci5„_i - C2Sn-2 +C 3 5„_3

h(-l)"

-2

C„_i^i+(-l)"

-1

tlCn.

(2·2)

We also have the duality formula l

(sn,b\

•••brkkk)

=

1

ifb\ l •••brkk

0

else.

=bn,

We will find it convenient to use the total symmetric functions hn, recursively defined by (-l)kcn-khk=0, 0 ζίαζη and also satisfying the recurrence relation •si = hi, s„ = nhn - (his„-i

+ h2sn-2 +

Η

l·

hn-isi).

(2.3)

These formulae combine to give the following remarkable result. Theorem 2.1. There is an isomorphism of (graded) Hopf algebras Φ: H*{BU)

H*(BU)·,

Φ (bn) = hn

under which Φ(pn) =

(—l)n_1in.

Let Λ be a Hopf algebra over a ring lk and let A* = Hom k (A, k) be its dual. Then there is a k-algebra homomorphism A*

End k (A);

α ι—* a· , where a •a =

ι{α')α",

with Σ a' a" denoting the coproduct on a. This gives a canonical action of A* on A. In the case where A = H*(BU), and A* = H*(BU), this action agrees with the cap product action of cohomology on homology. Thus, for u e H2m(BU) and x € Hln(BU) we have u • χ = uC\x.

Vertex operators in algebraic topology

9

Now given a k-algebra homomorphism φ: A —> A*, we have an induced action of A on itself given by a · χ = φ(α) · χ,

for a, χ € A .

Taking the case of A — H*(BU), we obtain the action u · χ = Φ(μ) Π χ. Proposition 2.2. We have the following formulae. 1) For a,b,x e H*(BU), (ab) χ = a • (b • χ). Hence • is a left action of H*(BU) on itself 2) For the primitives pn (n > 0), Pm ' Pn — n&m n. 3)

For the standard generators b^, bm • bn = hm Π bn =

bo = 1 if m = n, bn-m i/O ^ m < n, . 0

otherwise.

In terms of the generating function b(T), this formula is equivalent to b(X)-b{Y)

= (1 -

XY)~lHY)·

In the statement of part (2), the Kronecker symbol 8m = η + wt«" for 0 < d - 1. Now if wt un = 1 then wtu = n + l , so as η < d — 1 then wt υ < d. So in fact wt ν = d and η = d — 1 in this case. So if η < d — 2 then u" is quasi-primary of weight at least 2, so that deg(L(—1)"m") = η if un φ 0 by Lemma 3.1 and Theorem 2.3. Since ν has degree d >n + 1 and d&g{L{-l)d~xud~x) >d - 1 we see that L(-l)nu" = 0 for η < d - 2. So now ν — L(—\)d~lud~l with wtw^ - 1 = 1, forcing degud~l = 1 by Lemma 3.1. So u d ~ l e / i ( V ) , and the theorem follows. •

4. Theorem 1 First we prove

Lemma4.1. Ji(V) + (L(0) + L(—1))V c J(V). Proof It is enough to show that (L(0) + L(—V))v lies in J(V) for homogeneous ν e V. But we have L(0)u = (wtu)t; and (L(—\)v)n = — nvn-\, so taking η = wt(L(—l)n) — 1 = wt ν shows that o(L(0)v 4- L(— 1)υ) = 0 as required. • To begin the proof of Theorem 1, pick u e 7 ( V ) . As before we can write

m v=

n=0

(4.1)

The radical of a vertex operator algebra

23

where each « " i s semi-primary and where um φ 0. We prove by induction on m that ν lies in 7i(V) + (L(0) + L(-1))V. Suppose first that m = 0. Then ν = u m is semi-primary, whence the condition o(v) = 0 forces ν e V\ by Theorem 2.2. So in fact ν e J\ (V) in this case. In general, set χ = L{-\)m-lum and y = £|T=o L(-l)nun. Thus ν = L ( - l ) x + y . Now from Lemma 4.1 we have (L(0) + L(— 1))* € J(V), that is 0 = o(v)

= o(L(—l)x)

We easily check that

+ o(y)

= (m -

= -o(L(0)x)

\)L(-l)m~lum

+ o(y)

= o(y

-

L(0)x).

+ L(-l)m~lL(0)um

so that

m - 2

y-L(0)x

L(-l)nun

=

+ L(-l)m-l((m

-

\)um

+ L(0)um

+

um~x)

«=ο lies in J(V). Since L(0)um is semi-primary, we conclude by induction that y — L(0)x lies in 7i(V) + (L(0) + L ( - l ) ) V . But then the same is true of υ = y - L ( 0 ) x + (L(0) + L(—l))x. This completes the proof of the theorem. Remark. The referee has kindly pointed out that one can also deduce Theorem 1 from Theorem 3. We make some remarks about Heisenberg vertex operator algebras. These are constructed from a finite-dimensional abelian Lie algebra Η equipped with a non-degenerate, symmetric, bilinear form ( , ). One then forms the Z-graded affine Lie algebra L = / / ( g j Q f . f - ^ e C c

where [u tm, ν 1 be the rank of Φ, Ν = |Φ + |, and h the Coxeter number of Φ. It is well-known that we have 2N = Ih.

(2.1)

Any a € Φ + determines a partition of Φ + , namely Δ 0 (α) = {α} Δ!(α) = {β € Φ+ I (α, β) φ 0, β φ α)

(2.2)

Δ 2 (α) = { / 3 € Φ + I (α, β) = 0, β φ α}. Here, (·, ·) denotes the usual inner product associated with Φ, normalized so that (a, a) = 2 for a € Φ. We often write a ~ β in case β e Δι (a); of course ~ is a symmetric relation. We will define a certain Q-algebra A = Α(Φ). Additively it is a free abelian group with a distinguished basis consisting of elements t(a), u(a) with a e Φ + . Thus A has rank 2Ν. To define multiplication, note that if a ~ β with α, β e Φ + then there is a unique γ e Φ + Π Ζ (α, β) such that a ~ γ ~ β. We use this observation to define

= [J Σ "(«)) aeAi(ß) oteA\(ß) = ^ (δί(^)+Σ aeAi(j8)

+

-

(2-3)

w « ' 0»)+Σ aeAiiß)

+w(a) -

0»))

where γ {α, β) is the element of Φ + determined by a and β whenever a ~ β. But γ (α, β) ranges over Δι (β) as a does, so all terms in (2.3) cancel except for the t(ß) 's. Lemma 2.1 tells us that (2.3) is thus equal to ^ (81 (β) + (2h - A)t (β) + (2h - A)t (β)) = t(ß), that is t(ß)8 = t(ß). Similarly u(ß)8 = u(ß), and the proposition is proved. • We observe that the Q-span of the t(a) for a e Φ + is a subalgebra of A, which we denote by Γ(Φ). The same proof shows Lemma 2.3. Τ(Φ) has an identity, namely < = 2£T4 Σ

'·

α€Φ+

Now introduce a symmetric bilinear form (, ) on Λ(Φ) as follows: (f (or), t(a))

£),

.2 .A8

1:28(Dg),

2:64(A")

25.26.S6

16:1(D«),

18:6φ£), 2:28(A«)

4

A

7

59

2 .A8

64:1(A"),

ΑΒ7

7

«8

24.24.S4

30:8(A«)

7

7

«7 + 1

ES

8

58+2

A

9

8

59 + 1

24.27.A8

16: l ( D g ) ,

1:28(07),

Dß8

8

«8 + 1

24.27.(2454)

15:8(D7),

1:128(A^)

9

8

510

23.24.L3(2)

30:2(Ag),

8:28(Aß)

AY9

8

59+1

16:2(Ag),

1 : 2 8 (ΑΥΊ)

Y

A

D

A

A

l.S6 X 2 21+8.0+(2)

1.5g

32:2(A£), 1:120(^7),

1:36(A«)

l:64(Ag)

63:6(Ag) 1:135φ£) 1:128(A£)

38

C. Dong, Η. Li, G. Mason, S. P. Norton

Aß AY As a

D

D9 Aa ß

Aa Αβ9 As9 A€9

f A9 D a12

8 8 8 9 9 9 9 9 9 9 9 10 10

«9 «8 + 1 «7+2

«10 + 1 «9 + 1 0+ 12

AB A^12 A5 A€

£>24

A24

An Αβ15 AY12

16:9 (Αβ) 16:1 (Αβ), 30:8(A^) 10:9(A^) 15:2(D%), 4 : 2 8 ( D ß ) , 1:256(A«)

2 3 .2 8 .(3 2 .2S 4 ) 2 2 .2 6 .3 1+2 .2 2

8:9 (Dß), 1:256(A ß ) 14:18(Ag), 8:64(Ag) 8: 1(Aq), 30:2(Ag), 4:28(A£)

«9+1

1.24.L3(2) 2 2 .S 6 .2 1.32.254

«8 + 2 «8 + 2 4 +1

1.25.S5 1:10(Ag) 6 12:6(Ag), 27:4(Ag) l.[2 3] 2 2 .2 9 .(2 6 .3 1+2 .2 2 ) 7:18(D? 0 ), 4:64(Dß),

«10 +

1

«10

«ίο + 1 10 «16 10 4 + 1 Λ 11y 10 «12 Λ A11 10 «10 + 2 Λ A^ 10 10 «ίο + 1 Λ 10 10 «9 + 2 £>16 11 «16 + 1 Dß12 11 «12 + 1 A23 11 «24 A16 11 «16 + 1 z/ 4 a15 Aa12

2 3 .3 2 .2S 4 1.24.S4 I.S31S3 2 3 .2 8 .(2 4 .L 3 (2))

11 «+2+2 11 «12 + 1 11 «ίο + 2 11 «9 + 3 12 «24 + 1 12 «24 + 1 12 «16 + 2 12 4 + 4 12 «9+4

2

9

2 .2 .S6.2 2.24 .A 8 1.2 6 .3 1+2 .2 2 2.M12 l.[2 9 3] \.S6.2

6:10(Ag) 8:1 (A§), 8:9(Ag)

ß

1:512(A?,)

ß

3:10 (D ), 1:512 (A ) 6:140 (Aft), 4:448(A^) 4: 7:18(A" 0 ), 4:64(A*) 2:66(Ag)

2.210.Mi2 l.M 24 1.24.A8

7:4(A? 0 ), 10:4(A|), 3:24(Α^) 4:1(AJ), 3:10(Aj) 6:2(Aj), 12:9 (Αζ9) 3 : 1 4 0 ( D f 2 ) , 2:448(£>f0), 1:1024(A«5) l:66(Dß0), l:1024(Af,) 2:759(A" 5 ), 2:2576(Aß1) 2:1 (A«5), 3:140(A«2), 2:448(A?0)

l.[2 9 3 2 ]

3:2(A? 2 ), 6:18(A^), 2:64(A[ 0 )

I.A/12 l.S 6 x 2 1.32.254

2: l(A^j), l:66(Af 0 ) 2:2(Af 0 ), l:10(Af 0 ) 4:12(A[ 0 )

1.2π.Μ24

1:759(DI 6 ), l:2576(öf 2 ) 1:1(A 23 ), 1:759(A 16 ) 1:2(A 16 ), 1:140(AI 3 ), 1:448(A{,)

l.[2 4 3 2 ] 2.2 10 .(2 4 .A 8 )

I.M 24 1.24.A8 x 2 l.[2 12 3 3 ] l.L 3 (3)

1:24(AI3), 1 :256(Ajj) 1:13(Ajj)

Let us fix a Niemeier lattice L not equal to the Leech lattice, and let R denote the root lattice of the corresponding root system L2· So there are isometric embeddings VlR

V2L

A

Associative subalgebras of the Griess algebra and related topics

39

and correspondingly there are vertex operator algebra embeddings V

3i* ^

V

>2L -

F

A ^

^

0, ht~k acts like multiplication by ht~k and, when h is a root, h®tk acts like k times differentiation with respect to h. When η = 2, this means (x2, x2} — 2m2(x, x). In V f , m = I. Definition 2.2. The Symmetric Bilinear Form. Source: [FLM], p. 217. This form is associative with respect to the product (Section 3). We write Η for H\. The set of all g 2 and spans V2· (g2,h2)=2(g,h)2,

(2.2.1)

whence (pq, rs) = {p, r)(q, s) + {p, s){q, r), for p, q,r,s < ^

+

H o

(g2,x+)=0.

€ H.

(2.2.2) (2 2 3)

· ·

(2.2.4)

Notation 2.3. In addition, we have the distinguished Virasoro element ω and identity I := jω on V2 (see Section 3). If hi is a basis for Η and h* the dual basis, then ®=iE,-Mr·

A VOA related to Εg with automorphism group 0 + ( 10,2)

45

Remark 2.4. (g2,a>) = (g,g)

(2.4.1)

(8 2 Λ) = \ ( 8 , 8 )

(2.4.2)

(I, I) = dim(ff)/8

(2.4.3)

(ω, ω) = dim(tf )/2.

(2.4.4)

If {jc, 11 = 1, . . . £} is an ON basis, 1

1

(2.4.5) ι=0

ω= \ Σ

χ

ϊ ·


)· We ω define I μ '·= \ ο ) μ · If Μ and Ν are orthogonal sets, we have (omun = Μ + &>N • Define ω'Μ := ω — ω μ and Ι'Μ := I — Im- This element can be written as ω μ = \ Σί xf> where the Xj form an orthonormal basis of span(M). We have (com, com) =

46

R. L. Griess, Jr.

jdimspan(M) and (Ια/,Ιμ) = |dimspan(M). Also, (ω^,χγ) = (ω^,χ'γ'} = (ω, x'y'), where priming denotes orthogonal projection to span(M). Notation 4.2. ef := / λ τ := ^ [ λ 2 ± 4υ λ ], ex = e+, fx := e~. If a € Ζ or Z 2 , define ex%a to be or ej, as α ξ 0, 1 (mod 2), respectively; see (4.7). Also, let e'Xa= βχ,α+ι- We define tobe where a is λ, μ> incase μ is a vector in L, and a is [λ, μ], where [.,.] is the nonsingular bilinear form on Hom(L, {±1}) gotten from 2(.,.) by thinking of Hom(L, {±1}) as \ L j L and where λ is the character gotten by reducing the inner product with modulo 2. Finally, let q be the quadratic form on Hom(L, {±1}) gotten by reducing χ i-h- (X,X) modulo 2, for χ £ j L . Lemma 4.3. (i) The ef_ are idempotents. 16

(ω

(4>φ

=

1

128

0 0 ι (iii) (et,e*) = 128 "0

λ = μ; (λ, μ) = - 2 ; (λ, μ) = 0. λ = μ; (λ, μ) = - 2 ; (λ, μ) = 0.

Proof, (i) (e±)2 = - ^ [ 4 · 4λ 2 + 16λ2 ± 8 · 4 2 υ λ ] = e f . (ii) and (iii) follow trivially from (2.2). Notation 4.4. For finite X c L, define s(X) := Σχ€±χ/{±1} χ2 · Lemma 4.5. If X c L2, (a,s((x)) = (ω,ί(α)> = 4 α/κ/ (ω α ,ω α ) = j, whence s(a) = 8ωα — 161« and I« = j^a 2 . (ii) (ö)£7, ί(Φ£ 7 )) = (ω, ϊ(Φε 7 )) = 63, whence $(Φε7) = 18ωφ £? ; (iii) (ω,5(Φ£)8)) = 56, whence ^(Φβ 8 ) = 56ωφ£7Notation 4.7. For φ e Hom(L,{±l}), define f((p) := Σχ&ι2/{±\}ψ(λ·)νχ, u{