Lecture Notes in Mathematics

Edited by A. Dold and B. Eckmann Series: Mathematisches Institut der Universit~it Bonn Adviser: F. Hirzebruch

716

M. Scheunert

The Theory of Lie Superalgebras An Introduction ¢ Springer-Verlag Berlin Heidelberg New York 1979 Author

Manfred Scheunert Department of Physics University of Wuppertal D-5600 Wuppertal 1

AMS Subject Classifications (1970): 17 E05

ISBN 3-540-09256-0 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09256-0 Springer-Verlag NewYork Heidelberg Berlin

Library of Congress Cataloging in Publication Data ScheunerL M 1939- The theory of Lie superalgebras.(Lecture notes in mathematics; 716) Includes bibliographicalreferences and index. 1. Lie algebras. I. Title. II. Series: Lecture notes in mathematics(Berlin) ; 7i6. QA3.L28 no. 716 [QA252.3] 510'.8s [512'55] 79d5333

TMs work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publishel © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hernsbach/Bergstr. 2141/3140-.543210 To Irene PREFACE

The theory of Lie superalgebras (or, as they are also called, Z2-graded Lie algebras) has undergone a remarkable evolution during the last few years. At present the most important result in the theory seems to be the classification by V.G. Kac of the finite-dimensional simple Lie superalgebras over an algebraically closed field of characteristic zero. Our main objective is to give a self-contained and detailed presenta- tion of this classification. Thus we shall not presuppose any knowledge of the theory of Lie superalgebras, however, we assume that the reader is familiar with the standard theory of Lie algebras.

The present article has been written during the author's visit to the Dublin Institute for Advanced Studies, a stay which has been made pos- sible through a grant by the Deutsche Forschungsgemeinschaft. The kind hospitality at DIAS as well as the support by the DFG are gratefully acknowledged. Above all, thanks are due to V. Rittenberg; without his permanent interest and encouragement this work would have hardly been written.

Dublin Manfred Scheunert April, 1978 IABLE OF CONTENTS

Introduction 1

Chapter 0 Preparatory remarks 5

§1 Conventions 5

§2 Some general remarks on graded algebraic structures 6

Chapter I Formal constructions 12

§1 Definition and elementary properties of Lie superalgebras 12

§2 The enveloping algebra of a Lie superalgebra 19 I. Definition and some basic properties of the enveloping algebra 19 2. The supersymmetric algebra of a graded vector space 23 3. Filtration of the enveloping algebra and the Poincar~, Birk- hoff, Witt theorem 25 4. The enveloping algebra as a Hopf superalgebra 31

§3 Representations of Lie superalgebras 34 1. The connection between representations of L and U(L) 34 2. Canonical constructions with L-modules 37 A. Extension of the base field 37 B. The tensor product of graded L-modules 38 C. Representations in spaces of multilinear mappings 41 3. Invariants 45

§4 Induced and produced representations 51 I. Induced representations 52 2. Produced representations 54 3. Additional structures on produced modules: Filtration and multiplication 56 4. Some non-canonical constructions 60 5. The Guillemin, Sternberg realization theorem 62

Chapter II Simple Lie superalgebras 72

§1 Miscellanies on Z-graded and filtered Lie superalgebras 72 1. Some definitions concerning Z-graded Lie superalgebras and VIII

a criterion for two bitransitive Lie superalgebras to be isomorphic 72 2. Various results on transitive Lie superalgebras 77 3. Construction of two types of transitive Lie superalgebras 83 4. Filtration of Lie superalgebras 86

§2 Some general properties of simple Lie superalgebras 91 1. Some elementary results on simple Lie superalgebras 91 2. Discussion of the L6- module L] 96 3. Cartan subalgebras of a Lie superalgebra 108

§3 Lie superalgebras whose Killing form is non-degenerate 112 I. Some basic general results 112 2. The root space decomposition of a Lie superalgebra whose Killing form is non-degenerate 120

§4 The classical simple Lie superalgebras 124 i. The general linear Lie superalgebra pl(V) 124 2. The special linear Lie superalgebra spl(V) 127 3. Subalgebras of pl(V) which leave invariant a homogeneous non-degenerate bilinear form on V 129 A. The orthosymplectic Lie superalgebras 129 B. The Lie superalgebras b(n) 132 4. The (f,d) algebras of Gell-Mann, Michel, Radicati 133 5. Comments on the exceptional classical simple Lie superalgebras 134 6. The root space decomposition of the classical simple Lie superalgebras 136 §5 Classification-of the classical simple Lie superalgebras 140 I. A trivial preliminary remark 142 2. L~ is not simple, ad' is irreducible 143 3. L~ is not simple, ad' is not irreducible 148 4. L~ is simple 160 5. Extension of some classical simple Z-graded Lie superalgebras 163

§6 The Cartan Lie superalgebras 169 IX

1. The Lie superalgebra W(V) of superderivations of an exterior algebra 169 A. Definition and elementary properties of W(V) 169 B. W(V) as a sl(V)-module 173 C. W(V) as a universal transitive Z-graded Lie superalgebra 177 D. W(V) as a universal transitive filtered Lie superalgebra 181 2. The Lie superalgebras S(V) and S(V,t) 186 A. Elementary properties of S(V) 186 B. Elementary properties of S(V,t) , dimV even 189 C. Filtered Lie superalgebras whose associated Z-graded Lie superalgebra is isomorphic to S(n) 191 3. The Lie superalgebras H(~) and H(~) 194 A. Elementary properties of H(~) and H(~) 194 B. A characterization of the algebras H(~) and H(~) 197 C. Filtered Lie superalgebras whose associated Z - graded Lie superalgebra is isomorphic to H(~) or H(~) 202

§7 Classification of a special type of transitive Z-graded Lie superalgebras 208

§8 The main classification theorems 222

Chapter III A survey of some further developments 231

§I Superderivations of Clifford algebras and Lie superalgebras 231 I. Superderivations of a Clifford algebra 231 2. Superderivations of a Lie superalgebra 232

§2 A few remarks on nilpotent, solvable, and semi-simple Lie superalgebras 236 I. Nilpotent and solvable Lie superalgebras 236 2. Semi-simple Lie superalgebras 237

§3 Finite-dimensional representations of simple Lie superalgebras 239 i. Lie superalgebras all of whose finite-dimensional representations are completely reducible 239 2. Irreducible representations of simple Lie superalgebras 241 3. Generalized adjoint operations and star representations 243 X

Appendix 248

1. Notational conventions for reductive Lie algebras 248 2. Remarks on semi-simple Lie algebras and their representations 250 3. Special remarks on simple Lie algebras 252 4. A technical lemma 254 5. The index of a representation 258

References and foot-notes 262

Subject index 266