The Theory of Lie Superalgebras An Introduction
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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 - 5 4 0 - 0 9 2 5 6 - 0 Springer-Verlag Berlin Heidelberg NewYork ISBN 0 - 3 8 7 - 0 9 2 5 6 - 0 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in PublicationData ScheunerL M 1939- The theory of Lie superalgebras.(Lecture notes in mathematics;716) Includes bibliographicalreferencesand 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
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PREFACE The theory of Lie superalgebras (or, as they are also called, Z2-graded Lie algebras) has undergone a remarkable evolution during the l a s t few years. At present the most important r e s u l t in the theory seems to be the c l a s s i f i c a t i o n by V.G. Kac of the finite-dimensional
simple Lie su-
peralgebras over an algebraically closed f i e l d of c h a r a c t e r i s t i c zero. Our main objective is to give a self-contained and detailed presentation of this c l a s s i f i c a t i o n .
Thus we shall not presuppose any knowledge
of the theory of Lie superalgebras, however, we assume that the reader is f a m i l i a r with the standard theory of Lie algebras. The present a r t i c l e has been w r i t t e n during the author's v i s i t to the Dublin I n s t i t u t e for Advanced Studies, a stay which has been made possible through a grant by the Deutsche Forschungsgemeinschaft. The kind h o s p i t a l i t y at DIAS as well as the support by the DFG are g r a t e f u l l y acknowledged. Above a l l ,
thanks are due to V. Rittenberg; without his
permanent i n t e r e s t and encouragement t h i s work would have hardly been written. Dublin A p r i l , 1978
Manfred Scheunert
IABLE OF CONTENTS Introduction
1
Chapter 0
5
Preparatory remarks
§1 Conventions
5
§2 Some general remarks on graded algebraic structures
6
Chapter I
Formal constructions
12
§1 D e f i n i t i o n and elementary properties of Lie superalgebras
12
§2 The enveloping algebra of a Lie superalgebra
19
I. D e f i n i t i o n and some basic properties of the enveloping algebra
19
2. The supersymmetric algebra of a graded vector space
23
3. F i l t r a t i o n of the enveloping algebra
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