Lie Algebras and Lie Groups 1964 Lectures given at Harvard Universit
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Jean-Pierre Serre
Lie Algebras and Lie Groups 1964 Lectures given at Harvard University
Corrected 5th printing
~. Springer
Author Jean-Pierre Serre College de France 3, rue d'Ulm 75005 Paris, France
Mathematics Subject Classification (2000): 17B
2nd edition Originally (1st edition) published by: W. A. Benjamin, Inc., New York, 1965
Corrected 5th printing 2006 ISSN 0075-8434 ISBN-IO 3-540-55008-9 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-55008-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and Permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production: LE- TEX Jelonek, Schmidt & Vockler GbR, Leipzig Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
SPIN: 11530756
41/3142/YL
543210
Contents
Part I - Lie Algebras
1
Introduction
1
Chapter I. Lie Algebras: Definition and Examples............
2
Chapter II. Filtered Groups and Lie Algebras 1. Formulae on commutators 2. Filtration on a group 3. Integral filtrations of a group 4. Filtrations in GL( n) Exercises Chapter III. Universal Algebra of a Lie Algebra...... 1. Definition 2. Functorial properties 3. Symmetric algebra of a module 4. Filtration of U9 5. Diagonal map Exercises
6 6 7 8 9 10 ..
11 11 12 12 13 16 17
Chapter IV. Free Lie Algebras 1. Free magmas 2. Free algebra on X 3. Free Lie algebra on X 4. Relation with the free associative algebra on X 5. P. Hall families 6. Free groups 7. The Campbell-Hausdorff formula 8. Explicit formula..... Exercises
18 18 18 19 20 22 24 26 28 29
Chapter V. Nilpotent and Solvable Lie Algebras 1. Complements on g-modules 2. Nilpotent Lie algebras 3. Main theorems 3*. The group-theoretic analog of Engel's theorem........ 4. Solvable Lie algebras
31 31 32 33 35 35
VI
Contents
5. Main thoorem 5*. The group theoretic analog of Lie's theorem 6. ~as on endomorphisms 7. C8!"t8.ll'S criterion Exer-cises Chapter VI. Semisimple Lie Algebras 1. The radical 2. Semisimple Lie algebras 3. Complete reducibility 4. I..evi's thec>rem 5. Complete reducibility continued 6. Connection with co~pact Lie groups over R and C Exer-cises
36 38 40 42
43 44 44 44
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