Table Of Contentde Gruyter Expositions in Mathematics 7
Editors
O.K. Kegel, Albert-Ludwigs-Universität, Freiburg
V.P. Maslov, Academy of Sciences, Moscow
W.D. Neumann, Ohio State University, Columbus
R.O. Wells, Jr., Rice University, Houston
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Infinite Dimensional Lie Superalgebras
by
Yuri A. Bahturin
Alexander A. Mikhalev
Viktor M. Petrogradsky
Mikhail V. Zaicev
w
DE
G
Walter de Gruyter · Berlin · New York 1992
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Authors
Yu. A. Bahturin, A.A. Mikhalev, V.M. Petrogradsky
M.V. Zaicev Department of Mathematics
Department of Algebra Branch of Moscow State University in
Faculty of Mathematics and Mechanics Ulianovsk
Moscow State University 432700 Ulianovsk, Russia
119899 Moscow, Russia
7997 Mathematics Subject Classification: Primary: 15-02; 16-02; 17-02
Secondary: 15A66,15A75; 16R10,16S30,16W30; 17A70,17B37,17B65,17B70,17B81;20E06;
68Q40; 81R10, 81R50
© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Infinite dimensional Lie superalgebras / by Yuri A. Bahturin ...
[etal.].
p. cm. — (De Gruyter expositions in mathemat-
ics, ISSN 0938-6572 ; 7)
Includes bibliographical references and indexes.
ISBN 3-11-012974-4 (acid-free)
1. Lie algebras. I. Bakhturin, IU. A. II. Series.
QA252.3.I54 1992
512'.55-dc20 92-29650
CIP
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Infinite dimensional Lie superalgebras / by Yuri A. Bahturin
... - Berlin ; New York : de Gruyter, 1992
(De Gruyter expositions in mathematics ; 7)
ISBN 3-11-012974-4
NE: Bachturin, Jurij A.; GT
© 1992 by Walter de Gruyter & Co, D-1000 Berlin 30.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced or transmitted 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.
Typesetting: Asco Trade Typesetting Ltd., Hong Kong. Printing: Ratzlow Druck, Berlin.
Binding: Dieter Mikolai, Berlin. Cover design: Thomas Bonnie, Hamburg.
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Table of Contents
Preface vii
List of Symbols ix
Chapter 1
Basic facts about Lie superalgebras
§ 0. Some background 1
§ 1. Graded algebras 4
§ 2. Identical relations of graded algebras 22
Exercises 35
Comments to Chapter 1 37
Chapter 2
The structure of free Lie superalgebras
§ 1. The free colour Lie superalgebra, s-regular words and
monomials 39
§ 2. Bases of free colour Lie superalgebras 44
§3. The freeness of subalgebras and its corollaries 53
§ 4. Bases and subalgebras of free colour Lie p-superalgebras 69
§ 5. The lattice of finitely generated subalgebras 75
§ 6. Free colour Lie super-rings 78
Comments to Chapter 2 80
Chapter 3
Composition techniques in the theory of Lie superalgebras
§1. The Diamond Lemma for associative rings 81
§ 2. Universal enveloping algebras 84
§ 3. The Composition Lemma 95
§4. Free products with amalgamated subalgebra 105
Comments to Chapter 3 108
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vi Table of Contents
Chapter 4
Identities in enveloping algebras
§1. Main results 111
§2. Delta-sets 123
§ 3. Identities in enveloping algebras of nilpotent Lie superalgebras 129
§4. The case of characteristic zero 136
Comments to Chapter 4 144
Chapter 5
Irreducible representations of Lie superalgebras
§ 1. The Jacobson radical of universal enveloping algebras 147
§2. Dimensions of irreducible representations 152
§ 3. More on restricted enveloping algebras 160
§4. Examples 171
Comments to Chapter 5 173
Chapter 6
Finiteness conditions for colour Lie superalgebras
with identities
§ 1. Various types of finiteness conditions. Examples 175
§2. Maximal condition and Hopf property 180
§ 3. Sufficient conditions for residual finiteness 201
§4. Representability of Lie superalgebras by matrices 210
Comments to Chapter 6 236
Bibliography 237
Author Index 247
Subject Index 249
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Preface
In this book we consider some questions of the theory of Lie superalgebras
in the spirit of infinite-dimensional Lie algebras. The need for such a book is
motivated by the large number of results on this topic scattered through
many journals and, of course, by the importance of Lie superalgebras which
proved to be useful for quite a few areas of mathematics and physics. There
have appeared several books and surveys on Lie superalgebras, but, re-
flecting the initial stage of the development, they are devoted to special topics
of the theory without making an attempt of exposing general features.
In the first chapters of the book we are concerned with general questions of
the theory. After introducing the main definitions in Chapter 1 we consider
the notion of identity in a (colour) Lie superalgebra, the concept of a variety,
and prove some results on varieties of Lie superalgebras. In Chapters 2 and
3 we consider free Lie superalgebras. Chapter 2 is devoted mainly to the
question of finding linear bases in free Lie superalgebras i.e., essentially, to
the question of the canonical form for the elements of a Lie superalgebra in
terms of its generators. The bases obtained are similar to those well-known
in the case of ordinary Lie algebras, but they have interesting features of their
own. A part of Chapter 2 is devoted to the question of the freeness of sub-
algebras in free Lie superalgebras. The proof of the corresponding result,
which is an analogue of well-known theorems due to A.I. Shirshov and
E. Witt, gives an effective procedure for finding free generators of homo-
geneous subalgebras in Lie superalgebras in a number of important cases. In
Chapter 3 we discuss Lie superalgebras given in terms of generators and
defining relations. A number of general results and techniques developed
are applied to various constructions such as universal enveloping algebras,
free products of algebras with amalgamation and some others. The same
approach enables us to solve some algorithmic problems in this field.
Chapters 4 and 5 deal with a topic of a more specific nature which is
nevertheless of considerable importance, namely the theory of universal
enveloping algebras for Lie superalgebras. These are graded associative
algebras from which Lie superalgebras can be obtained by introducing the
operation of the supercommutator. The importance of universal enveloping
algebras lies in the fact that any graded representation of a Lie superalgebra
is a representation of this associative algebra; and the converse is also true.
Thus, the methods of associative rings used and developed here are of great
importance for the representation theory of Lie superalgebras and give a
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viii Preface
non-standard approach to results in the area previously developed by other
methods. In Chapter 4 we study the question of the structure of the universal
enveloping algebra. In particular, we consider conditions under which such
an algebra satisfies a non-trivial polynomial identity. In fact, we consider
these questions in the setting of restricted enveloping algebras, i.e., here the
restricted Lie superalgebras are involved. Some of the results proved here
provide solutions to some well-known problems. In Chapter 5 we give an
application of the results about identities to the question of the boundedness
of dimensions of irreducible representations for a Lie superalgebra over fields
of any characteristic. Of independent interest are the results on the Jacobson
radical as well as those on von Neumann regularity of enveloping algebras.
In the concluding Chapter 6 we consider the questions of residual
fmiteness, of representability by matrices, and of some other fmiteness condi-
tions for Lie superalgebras. Here we use some techniques developed earlier
by us; also we suggest some new approaches which are necessary for the
solution of the problems arising here.
The book is organized as follows. Each chapter is further divided into
sections and these latter into subsections. The three digit numbering k.l.m.
means Subsection m in Section § 1 of Chapter k. We write § k.l for Section
§ 1 in Chapter k. We omit k. or k.l. if we are within Chapter k or Section § k.l.
We do not make any historical or bibliographical remarks in this introduc-
tion; at the end of each chapter there is a section of comments on these
matters. The bibliography does not strive to embrace all of the literature on
Lie superalgebras; though it would be desirable to have a complete list of
publications of such an important area of mathematics.
This book owes its existence to the creative atmosphere at the Department
of Algebra at Moscow University, a small unrivalled algebraic universe. We
would like to thank our teachers and colleagues there for stimulating our
research and shaping our view of mathematics. The first author gratefully
acknowledges support by DAAD.
Moscow, May 1992 Yu.A. Bahturin
A.A. Mikhalev
V.M. Petrogradsky
M.V. Zaicev
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List of Symbols
M 2 A(X) 39
End M 2 A(X\ 39
K
ad 2 F(X) 39
^"(M, N) 2 F(X) 39
g
[S, T] 3 l+X) 39
L(n} 3 40
L" 3 M 39
υ (L) 3,85 ψ 43
π
&(V) 6 (μ η) 43
gr A 7 Ψ(α) 43
9(R) 8 π: L(X) -+ [Χ] 47
gl(n,m) 10 W(y. α ) 48
sl(n, m) 11 SWioc α ) 49
λ
Πλ /κ
12 MW) ^ 49,70
6
LxM 13 ^(w) 49,70
ε 14 ad' 53
[]Λ 15 Z(x) 54
ε
x[pl 18 W (z) 54,71
33 23 Z 54
Ft*,») 23 W 54
var^ 24 a c>· b 54
fi(K,G,e) 24 ω: S -> L(X) 57
93 (G, ε) 25 PS[JSf] 69
s
26 Ψ:Ρ5[ΛΓ] -^S(AT) 69
y
Sym(m) 26 FZ(x) 71
M(i/) 26 PZ(x) 71
M(AT) 27 «(L) 87
A wr L 28 ® 88
#(L) 29 17" 89
29 gr U (L) 89
Η^,...,^,!^,...,^)
H(L, i) 29 ^(H) 92
QJ2 34 φ:Χ-+Ν 96
39 a 96
\·^ /
T~V V\ 39 Π α°Η 105
α
S(Jf) 39 aeT
r.
SW. 39 112
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List of Symbols
121 D m(L) 157
a
..,*„, yi,...,yj 122 D.(L) 157
123 D(L) 157
124 1 (T) 164
A
124 -ι Τ)Λ 164
124 /(L) 164
A(L) 124 r(L) 164
Rad(K) 130 Ass( ) 211
132 »!» 233
2
132 9l 233
k
149 31 233
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