Table Of ContentCOHOMOLOGY OF
INFINITE-DIMENSIONAL
LIE ALGEBRAS
CONTEMPORARY SOVIET MATHEMATICS
Series Editor: Revaz Gamkrelidze, Steklov Institute, Moscow, USSR
COHOMOLOGY OF INFINITE-DIMENSIONAL LIE ALGEBRAS
D. B. Fuks
LINEAR DIFFERENTIAL EQUATIONS OF PRINCIPAL TYPE
Yu. V. Egorov
THEORY OF SOLITONS: The Inverse Scattering Method
S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov
TOPICS IN MODERN MATHEMATICS: Petrovskii Seminar No.5
Edited by O. A. Oleinik
COHOMOLOGY OF
INFINITE·DIMENSIONAL
LIE ALGEBRAS
D. B. Fuks
Moscow State University
Moscow, USSR
Translated from Russian by
A. B. Sosinskii
CONSULTANTS BUREAU • NEW YORK AND LONDON
Library of Congress Cataloging in Publication Data
Fuks, D. B.
Cohomology of infinite·dimensional Lie algebras.
(Contemporary Soviet mathematics)
Translation of: Kogomologii beskonechnomernykh algebr Li.
Includes bibliographical references and index.
I. Lie algeoras. I. Title. II. Series.
QA252.3.F8513 1986 512'.55 86·25298
ISBN 978-1-4684-8767-1 ISBN 978-1-4684-8765-7 (eBook)
DOI 10.1007/978-1-4684-8765-7
This translation is published under an agreement with YAAP, the Copyright Agency
of the USSR
© 1986 Consultants Bureau, New York
Softcover reprint of the hardcover 1s t edition 1986
A Division of Plenum Publishing Corporation
233 Spring Street, New York, N.Y. 10013
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
I dedicate this book to Israel Gelfand
on the occasion of this seventieth
birthday
Foreword
There is no question that the cohomology of infinite
dimensional Lie algebras deserves a brief and separate mono
graph. This subject is not cover~d by any of the tradition
al branches of mathematics and is characterized by relative
ly elementary proofs and varied application. Moreover, the
subject matter is widely scattered in various research
papers or exists only in verbal form.
The theory of infinite-dimensional Lie algebras differs
markedly from the theory of finite-dimensional Lie algebras
in that the latter possesses powerful classification theo
rems, which usually allow one to "recognize" any finite
dimensional Lie algebra (over the field of complex or real
numbers), i.e., find it in some list. There are classifica
tion theorems in the theory of infinite-dimensional Lie al
gebras as well, but they are encumbered by strong restric
tions of a technical character. These theorems are useful
mainly because they yield a considerable supply of interest
ing examples. We begin with a list of such examples, and
further direct our main efforts to their study.
vii
viii FOREWORD
The work consists of three chapters. After the brief
Chapter 1 ("General Theory"), we begin the systematic com
putation of the cohomology of infinite-dimensional Lie al
gebras in Chapter 2. The main results of this chapter con
cern the algebras of formal and smooth vector fields, cur
rent algebras, and Kac-Moody algebras. (The first and last
sections of this chapter deal with another topic: the co
homology of finite-dimensional Lie algebras and the cohomol
ogy of Lie superalgebras; the latter, in their methods, re
sults, and applications, are fairly close to the homology of
infinite-dimensional Lie algebras.) The concluding chapter
is devoted to applications. These applications comprise the
characteristic classes of foliations, combinatorial identi
ties known as the Macdonald identities, invariant differen
tial operators, cohomology, and, in particular, central ex
tensions of Lie groups and cohomology operations in cobord
ism theory.
The chapters are divided into sections, and the sections
into subsections. Some of these subsections are further di
vided into smaller subsections, denoted by capital letters.
When we refer to a section from another chapter, we add the
number of the chapter to the number of the section; when re
ferring to a subsection from another chapter, we add the
numbers of the chapter and section to the number of the sub
section. Thus, Sub-subsection D of Subsection 3 of Section
2 of Chapter 1 is written 1.2.3D; in Chapter I this notation
is abbreviated to 2.3D; in §1.2, to 3D; and in Subsection
1. 2. 3, to D.
It is by no means necessary to read the book in sequence:
it is possible to reach any specific result avoiding irrele-
FOREWORD ix
vant facts. For example, the reader who is interested only
in characteristic classes of foliations may limit himself
to §§1.1-1.3, Subsections 1.5.1, 1.5.2, and §§2.l, 2.2, and
3.1. The reader who would like to become acquainted with
the Macdonald identities may do so by reading §§1.1-1.3 and
Subsections 1.5.3, 2.S.l, and 3.2.1-3.2.3; as additional ma
terial, he may leaf through Subsections 2.s.3A-B and 3.2.4.
I tried to limit lacunas in proofs to a minimum, but
did not succeed in avoiding them altogether. In the corres
ponding places references are supplied, but the fact that
completely proved theorems are interspersed with partially
proved theorems might be a certain inconvenience for the
reader. To avoid confusion, I shall say at once that the
main lacunas in the proofs are contained in §§2.3, 3.3, and
3.5. In addition, many sections are concluded by results
not incorporated in the main text and given without proof;
these results are usually grouped together in a separate sub
section (such as Subsections 2.1.5, 2.2.7, 2.4.3B, 2.5.3,
2.6.3, 3.2.4, and 3.4.3). Finally, in a number of places I
omit proofs or parts of proofs because they are similar to
other proofs presented in this book; recovering these omis
sions should be viewed as an exercise for the reader.
The standard graduate course in mathematics is suffi
cient for understanding the text. Systematically we use
facts from the classical theory of groups and Lie algebras,
homology algebra, and topology, but, as a rule, these are
the simplest definitions and theorems presented on the first
pages of textbooks in these subjects. An exception is §J.5,
the last section of the book, where some serious knowledge
of topology is required.
x FOREWORD
One can say without exaggeration that the book's sub
ject matter is due to Israel Gelfand, to whom I dedicate
this book. At the outset, he was the initiator and first
enthusiast in the computation of the cohomology of infinite
dimensional Lie algebras. He and his pupils are the authors
of many key results in this field. I do not think there is
any single theorem in this work which was not reported at
his seminar.
I am grateful to R. V. Garnkrelidze for suggesting that
I write this book. I am grateful to V. M. Bukhshtaber,
D. A. Leites, and especially B. L. Feigen for assistance. I
also thank my colleague A. B. Sosinskii for his competent
translation, Mrs. Ira Bychkova for her superior typing, and
my wife Ira for her patience and help in preparing the manu
script.
D. Fuks
Contents
Chapter 1. GENERAL THEORY
§l. Lie algebras.............................. 1
§2. Modules... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
§3. Cohomology and homology................... 15
§4. Principal algebraic interpretations of
cohomology ................................ 30
§5. Main computational methods... .............. 40
§6. Lie superalgebras 49
Chapter 2. COMPUTATIONS
§l. Computations for finite-dimensional Lie
algebras .................................. 61
§2. Computations for Lie algebras of formal
vector fields. General results...... ..... 75
§3. Computations for Lie algebras of formal
vector fields on the line ...... ........... 119
§4. Computations for Lie algebras of smooth
vector fields ............................. 140
§5. Computations for current algebras ......... 177
§6. Computations for Lie superalgebras ........ 201
Chapter 3. APPLICATIONS
§l. Characteristic classes of foliations ...... 209
§2. Combinatorial identities.................. 257
§3. Invariant differential operators...... .... 271
xi
