Lectures on Infinite Dimensional Lie Algebra

lectures en

Infinite-Dimensional Lie Algebra

Minoru Wakimoto

World Scientific

Lectures on

Infinite-Dimensional Lie Algebra

lectures on

Infinite-Dimensional Lie Algebra

Minora Wakimoto Kyushu University, Japan

>@ World Scientific ll

New Jersey • London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

LECTURES ON INFINITE-DIMENSIONAL LDZ ALGEBRA Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4128-3 ISBN 981-02-4129-1 (pbk)

Printed in Singapore by Uto-Print

To the memory of my parents

Preface The theory of integrable representations of affine Lie algebras is of great importance in connection with various areas of mathematics and mathematical physics. There are well-written books by Victor G. Kac [100], [115] and [101] where a beautiful exposition on the theory of Kac-Moody Lie algebras, Virasoro algebra and vertex algebras is displayed with detailed explanations. We begin with a brief sketch on the structure and representations of affine Lie algebras, which is used in this book, and then proceed to subjects of further interest. The Weyl-Kac type character formula for so-called generalized Kac-Moody Lie algebras and superalgebras and the modular properties of characters of integrable representations of affine Lie algebras are explained in [170], and readers are expected to refer to [100] or [170] for these subjects. The aim of this book is to give a detailed exposition on some important topics in the theory of affine Lie algebras, which have escaped from [170] for want of space. One of the main topics treated in this book is the theory of principal admissible representations which were discovered in the study of modular properties and conformal properties. The characters of principal admissible representations are closely related with characters of representations of W-algebras. We describe in detail their modular transformation and calculate fusion coefficients of W-algebras by making use of modular transformation of characters of integrable and principal admissible representations of affine Lie algebras. I learned the theory of quantized Drinfeld-Sokolov reduction from Edward Prenkel. I thank Akihiro Tsuchiya for encouragement while writing this book, and Lakshmi Narayanan for kind suggestions while completing this manuscript. Many of the materials in this book are obtained through joint research with Victor G. Kac. The collabolation with him has been so illuminating and fascinating to me. I would like to extend my heartleft thanks to Victor G. Kac on this occasion. I also wish to express my gratitude to my wife Yasuko. Fukuoka, Summer 2001

vn

Contents Preface

vii

1 Preliminaries on Affine Lie Algebras 1.1 Affine Lie Algebras 1.2 Extended Affine Weyl Group 1.3 Some Formulas for Finite-Dimensional Simple Lie Algebras

1 1 18 25

2 Characters of Integrable Representations 2.1 Weyl-Kac Character Formula 2.2 Specialized Characters 2.3 Product Expression of Characters 2.4 Modular Transformation

31 31 42 49 58

3 Principal Admissible Weights 3.1 Admissible Weights 3.2 Principal Admissible Weights 3.2.1 Prinicipal Admissible Weights with H\ = 5( u ) . 3.2.2 Principal Admissible Weights with n A = y(S(u)) 3.3 Characters of Principal Admissible Representations . 3.4 Parametrization of Principal Admissible Weights . . 3.5 Modular Transformation

. . • • . . . .

75 76 83 83 89 95 100 108

4 Residue of Principal Admissible Characters 4.1 Non-Degenerate Principal Admissible Weights 4.2 Modular Transformation of Residue 4.3 Fusion Coefficients 4.3.1 The Case g c d ( p - p ' , | J | ) = l 4.3.2 T h e C a s e g c d ( V , | J | ) = l 4.3.3 General Case

113 113 118 128 131 134 143

5 Characters of Affine Orbifolds 5.1 Characters of Finite Groups 5.2 Fusion Datum 5.3 Characters of Affine Orbifolds

153 154 162 168

IX

x

Contents

6 Operator Calculus 6.1 Operator Products 6.2 Boson-Fermion Correspondence

175 175 230

7 Branching Functions 7.1 Virasoro Modules 7.2 Virasoro Modules of Central Charge - ^ 7.3 Branching Functions 7.4 Tensor Product Decomposition

239 239 254 263 272

8 W-algebra 8.1 Free Fermionic Fields ip(z) and ip*(z) 8.2 Free Fermionic Fields cj>(z) and <j>*{z) 8.3 Ghost Field Associated to a Simple Lie Algebra 8.4 BRST Complex 8.5 Euler-Poincare Characteristics

287 287 291 294 297 301

9 Vertex Representations for Affine Lie Algebras 9.1 Simple Examples of Vertex Operators 9.1.1 The Space C ^ s j e N ] 9.1.2 The Space C[XJ;J e N o d d ] 9.2 Basic Representations of sl(2, C) 9.2.1 Homogeneous Picture 9.2.2 Principal Picture 9.3 Construction of Basic Representation 9.3.1 Homogeneous Picture 9.3.2 Principal Picture

321 321 322 327 336 336 345 351 352 366

10 Soliton Equations 10.1 Hirota Bilinear Differential Operators 10.2 KdV Equation and Hirota Bilinear Differential Equations . 10.3 Hirota Equations Associated to the Basic Representation . 10.3.1 Homogeneous Case 10.3.2 Principal Picture 10.4 Non-Linear Schrodinger Equations

385 385 387 392 394 404 414

Bibliography

429

Index

441

Chapter 1

Preliminaries on Affine Lie Algebras In this chapter, we give a brief and quick review of basic and fundamental notions and results on affine Lie algebras, which will be used in this book. For more detail discussions and proofs on the materials, readers are expected to refer to the Kac's book [100]. 1.1

Affine Lie Algebras

In this book, a vector space is always a complex vector space unless otherwise stated, and the terminology "linear" is used to mean the complex linear. The vector space may be infinite-dimensional, but the cardinality "infinite" always means the countable-infinity. For complex vector spaces V and W, Hom(V, W) denotes the vector space of all linear maps from V to W. A matrix A = (oij)»,j=i,— ,T» £ Mn(Z) satisfying the condition (CI)

au = 2

( t = l , ••-,»),

(C2)

atj < 0

if i / j ,

(C3)

an ^ 0 «=> oj-i ^ 0

is called a generalized Cartan matrix (GCM) or a Kac-Moody-Cartan matrix. Generalized Cartan matrices A = (a»j)i,j=i,-.. , n a n ( i A1 = (aij)»,j=i,— ,n are equivalent if a'ij =

a

|X) (*>4) : O

«i-3

«2

Q-

E

o-

cc=o a^-2

a 2 ) £

(2)

a0

o-

ai

O

a2

2 and A + A%\

M : ^ Q / [Q

(1.14)

A detail discussion on the structure of the Weyl group of an affine Lie algebra is described in Chapter 6 of [100]. For an element a G Yli=o Gat, we define ta G GL(h*) (or G-L(h)) by taX

:= X+(X\6)a-i^-(X\8)

+ (X\a)\s

(X G h*),

(1.15)

called the translation operator. It is easy to check the properties i Q °4 oo

(1.18b)

and e is the identity element (see Proposition 3.13 in [100]). We put Are

:= W(U),

Aim

and

:= A - A r e .

A root a belonging to A r e (resp. A l m ) is called a real (resp. imaginary) root. Another characterization of real or imaginary roots, for a Kac-Moody Lie algebra, is that Are Aim

:= :=

{a e A ; (a\a) > 0}, {a G A ; (a| a) < 0}.

In the case of an affine Lie algebra, imaginary roots are described as follows: Aim

=

{aeA;

(a| a) = 0} =

{n6 ; n G Z - {0}},

and the multiplicity mult(n^) := dimgns of an imaginary root n6 is given, for A = X%\ as follows: mult(n(5) = ^

if r = 1,

(1.19)

and if n grZ mult(n^)

N

-(.

if n G rZ,

(1.20)

1.1. Affine Lie Algebras

13

when r > 2. For each a G A r e , the element av

:=

2a (a|a)

(1.21)

is called a real coroot. Note that (toa) v

:=

w(av)

aeAre).

(w€W,

(1.22)

We put AVre

AVre

:=

w(nv)

:=

Av"n

-

{av ; a e A ™ } , £Z>0atv

=

{av ; a 6

Elements in A ^ r e are called "positive real coroots". We note that the map Are 3 a t—- av G A V r e

(1.23)

is bijective, but is not linear unless A is symmetric. This is just the reason why we make use of coroots but not of roots for the description of admissible weights in Chapter 2. Actually, for a, (3 G A r e and n e Z , the condition that a + n{3 is a root does not necessarily imply that a v + n/3v is a coroot, and vice versa. A real positive coroot a v is called simple if it is not a sum of two positive coroots. Simple roots and simple coroots correspond to each other under the map (1.23), and so the consistency of the notation 0%. Thus

n v = {a? ;

0
jW

—

0j(\)+mAj.

2) In the formula of 1), we let A = Aj and replace Ui\ ® U < + Ui ® [X> U ' ] ) 1=1

d

=

E , Biiw3

E 7 >kw d

d

d

y^ y ^ AjjMj ® M*+y^ y~] B^m ® ^ d

d

i=i j = i

And the rests are proved in the same way.

•

The above 2) implies that the element Q, called the Casimir element, commutes with all elements in g since d

[x, J2] = ^[x, i=l

d

UiU1} = ^2([x, Ui]ul + Ui[x, u1]) = 0. i=l

So the Casimir element £1 is a scalar operator on a highest weight g-module. It is sometimes convenient to write it in terms of root vectors as follows. Let A be the set of all roots of g, and A + the set of all positive roots. We identify the Cartan subalgebra f) with its dual space h* by using the inner

28

1. Preliminaries

product ( | )• Let Hi,--- ,Hi be an orthonormal basis of f) and, for each a 6 A, choose an element ea € ga such that \ya\&—a)

J-?

This condition implies that [6a, 6_aJ

=

Oi.

Then the Casimir element 12 is written in terms of this basis as follows: i i=l

a€A

Rewriting this formula into the following form is also quite useful: £

12 = ^2^iHi+ i=l

5Z e«e- 0 | ,

2. Integrable

36

Representations

This proposition says that, in this case, all maximal weights are W-conjugate to each other. Prom this proposition and (2.7), one obtains another expression of characters of level 1 integrable modules: Corollary 2.1.6. Let g be a simply-laced or twisted affine Lie algebra, and A e P+ be of level 1. Then

chL(A) = n

• Y i & h + a ^2

_ j-S)mult{nS)

n = l *>

+(A|a)

^•

aSM

'

For an affine Cartan matrix A of type X^ , w e define the function oo

GA(q) := JJ( 1 -9 B ) m u l t ( n ' ) -

(2-9)

n=l

One can write this function in more explicit form, since the multiplicities of imaginary roots n6 are known by (1.19) and (1.20). In particular in the case when r > 2, the function GA{q) is computed as follows:

GA(q) =

(i-^-n^1-'™)

n

i—l

n£N-rN

0 }.

2.1. Weyl-Kac Character Formula Thus the set of weights of L(AQ is given as follows: P(A 0 ) =

{A0 + jax + k6 ; j,keZ,

-j2}.

k