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de 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 Unauthenticated Download Date | 6/20/16 5:51 PM Unauthenticated Download Date | 6/20/16 5:51 PM 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 Unauthenticated Download Date | 6/20/16 5:51 PM 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 mathematics, 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. Unauthenticated Download Date | 6/20/16 5:51 PM 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 Unauthenticated Download Date | 6/20/16 5:51 PM 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 Unauthenticated Download Date | 6/20/16 5:51 PM 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 subalgebras 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 Unauthenticated Download Date | 6/20/16 5:51 PM 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 conditions 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 Unauthenticated Download Date | 6/20/16 5:51 PM 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 Unauthenticated Download Date | 6/20/16 5:51 PM 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 Unauthenticated Download Date | 6/20/16 5:51 PM