Lie Algebras
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127 Ian Stewart University of Warwick, Coventry/England
Lie Algebras
Springer-Verlag Berlin . Heidelberg . NewYork 1970
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zurich Series: Mathematics Institute, University of Warwick Adviser: D. B. A. Epstein
127 Ian Stewart University of Warwick, Coventry/England
Lie Algebras
Springer-Verlag Berlin . Heidelberg . NewYork 1970
This 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 § :>4 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 he determined by agreement with the publisher. © by Springer-Verlag BerUn ·Htidefbetg 19.70. Library of Congress Catalog Card Number 73-Umo. Printed in Germany. Tide No. 3283.
PREFACE These notes are based on an M.Sc. course which I gave at Warwick during the Autumn of 1969.
The material divides into two largely
disjoint sections. Part 1 is a relatively direct exposition of the classification theorem for finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero. entirely algebraic, with no appeal to analysis: groups are not mentioned.
The treatment is
connections with Lie
The main source for this section is a
lecture course given at Warwick in 1966 by Prof. R.W.Carter, which itself derived from lectures of Philip Hall at Cambridge.
Virtually
all of the material in Part 1 13 included with the classification theorem in view, so that I have not explored a number of interesting side-issues (such as
splitting theorem, or representations of
semisimple algebras).
This approach was dictated by two considerations:
time;
and a desire to exhibit the bare bones of the proof without
extraneous matter. Part 2 is drawn from two sources:
a paper pUblished by Brian
Hartley in 1966, and my Ph.D. thesis (1969).
The object here is to
investigate the structure of infinite-dimensional Lie algebras in the spirit of infinite group theory.
In particular we consider the Lie
analogue of a subnormal subgroup (which we call a sub ideal) and
.
consider the connections between the sub ideals of a Lie algebra and the structure of the algebra as a whole.
This branch of the subject
is at an early stage of development, as is underlined by the appearance in the text of a number of open questions.
r am grateful inspirationwise to Brian Hartley and Roger Carter, and perspirationwise to Sue Elworthy who typed the manuscript. Ian Stewart
CONTENTS Ch:.:onter Q
Basic Definitions
1
CLASSICAL THEORY OF FINITE DIMENSIONAL ALGEBRAS
PART ONE:
1
Representations of Nilpotent Algebras
10
2
Cartan SUbalgebras
15
3
The Killing Form
19
4
The Cartan Decomposition
24
5
Systems of Fundamental Roots
28
6
Dy
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