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A.L. Onishchik E.B. Vinberg Lie Groups and Algebraic Groups A A.L. Onishchik E.B. Vinberg Lie Groups and Algebraic Groups A. L.Onishchik E. B. Vinberg Lie Groups and Algebraic Groups Translated from the Russian by D. A. Leites Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Arkadij L. Onishchik Department of Mathematics, Yaroslavl University 150000 Yaroslavl, USSR Ernest B. Vinberg Chair of Algebra, Moscow University 119899 Moscow, USSR Title of the Russian edition: Seminar po gruppam Li i algebraicheskim gruppam, Publisher Nauka, Moscow 1988 This volume is part ofthe Springer Series in Soviet Mathematics Advisers: L.D. Faddeev (Leningrad), R.V. Gamkrelidze (Moscow) Mathematics Subject Classification (1980): 22EXX, 17BXX, 20GXX, 81-XX ISBN-13:978-3-642-74336-8 e-ISBN-13:978-3-642-74334-4 DOl: 10.1007/978-3-642-74334-4 Library of Congress Cataloging-in-Publication Data Vinberg, E.B. (Ernest Borisovich) [Seminar po gruppam Li i algebraicheskim gruppam. English] Lie groups and algebraic groups/A.L. Onishchik, E.B. Vinberg; translated from the Russian by D.A. Leites. p. cm.~(Springer series in Soviet mathematics) Translation of: Seminar po gruppam Li i algebraicheskim gruppam. Includes bibliographical references. ISBN-13:978-3-642-74336-8 (alk. paper) 1. Lie groups~Congresses. 2. Linear algebraic groups-Congresses. I. Onishchik, A.L. II. Title. III. Series. QA387.V5613 1990 512'.55-dc20 89-21693 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 microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1990 Softcover reprint of the hardcover 1st edition 1990 Typesetting: Asco Trade Typesetting Ltd., Hong Kong 2141/3140-543210~Printed on acid-free paper Preface This book is based on the notes of the authors' seminar on algebraic and Lie groups held at the Department of Mechanics and Mathematics of Moscow University in 1967/68. Our guiding idea was to present in the most economic way the theory of semisimple Lie groups on the basis of the theory of algebraic groups. Our main sources were A. Borel's paper [34], C. ChevalIey's seminar [14], seminar "Sophus Lie" [15] and monographs by C. Chevalley [4], N. Jacobson [9] and J-P. Serre [16, 17]. In preparing this book we have completely rearranged these notes and added two new chapters: "Lie groups" and "Real semisimple Lie groups". Several traditional topics of Lie algebra theory, however, are left entirely disregarded, e.g. universal enveloping algebras, characters of linear representations and (co)homology of Lie algebras. A distinctive feature of this book is that almost all the material is presented as a sequence of problems, as it had been in the first draft of the seminar's notes. We believe that solving these problems may help the reader to feel the seminar's atmosphere and master the theory. Nevertheless, all the non-trivial ideas, and sometimes solutions, are contained in hints given at the end of each section. The proofs of certain theorems, which we consider more difficult, are given directly in the main text. The book also contains exercises, the majority of which are an essential complement to the main contents. As a rule, the generally accepted terminology and notation is used. Neverthe­ less, two essential deviations should be mentioned. Firstly, we use the phrase tangent algebra of a Lie group for the Lie algebra associated with this group, with a view to emphasizing the construction of this Lie algebra as the tangent vector space to the Lie group. Secondly, in contrast to some monographs and textbooks, we call a Lie subgroup of a Lie group any of its subgroups which is an embedded (and necessarily closed) submanifold, while an immersed submanifold endowed with the structure of a Lie group is called a virtual Lie subgroup. The reader is required to have linear algebra, the basics of group and ring theory and topology (including the notion of fundamental group) and to be acquainted with the main concepts of the theory of differentiable manifolds. Numbering of subsections, formulas, theorems, etc. is performed inside each section and sections are numbered inside a chapter. In references we generally use triple numbering: for instance, Problem 2.3.17 refers to Problem 17 of § 3, Chapter 2. However, we skip the number of a chapter (or a section) in references inside of it. The last chapter is not divided into sections but in references is considered consisting of one section: § 1. VI Preface In compiling the first draft of seminar's notes we enjoyed the help provided by E.M. Andreyev, V.G. Kac, B.N. Kimelfeld and A.C. Tolpygo. In computing the decompositions of products of irreducible representations (Table 5) B.N. Kimelfeld, B.O. Makarevich, V.L. Popov and A.G. Elashvili took part. Besides, we would like to point out that certain nice proofs were the result of seminar's workout. We are grateful to D.A. Leites thanks to whose insistence and help this book has been written. The Translator's Preface In my 20 years of work in mathematics, I have never met a Soviet mathe­ matician personally involved in any aspect of representation theory who would not refer to the rotaprint notes of the Seminar on algebraic groups and Lie groups conducted by A. Onishchik and E. Vinberg with the participation of A. Elashivili, V. Kac, B. Kimelfeld, and A. Tolpygo. The notes had been published in 1969 by Moscow University in a meager number of 200 copies. Ten years later A. Onishchik and E. Vinberg rewrote the notes and considerably enlarged them. This is a translation of the enlarged version of the notes; its abridged variant was issued in Russian in 1988. The reader might wonder why one should have the book: why not Bourbaki's book, or S. Helgason's, or one of the excellent (text) books, say, by J. Humphreys or C. Jantzen. Here are some important reasons why: - Nowhere are the basics of the Lie group theory so clearly and concisely expressed. - This is the only book where the theory of semisimple Lie groups is based systematically on the technique of algebraic groups (an idea that goes back to Chevalley and is partly realized in his 3 volumes on the theory of Lie groups (1946, 1951, 1955)). - Nowhere is the theory of real semisimple finite-dimensional Lie groups (their classification and representation theory included) expressed with such lucidity and in such detail. - The unconventional style-the book is written as a string of problems­ makes it useful as a reference to physicists (or anyone else too lazy to bother with the proof when a formulation suffices) whereas those interested in proofs will find either complete solutions or hints which should be ample help. (They were ample for some schoolboys, bright boys I must admit, at a specialized mathe­ matical school in Moscow.) - The reference chapter contains tables invaluable for anybody who actually has to compute something pertaining to representations, e.g. the table of decom­ positions into irreducible components of tensor products of some common representations, which is really unique. The authors managed to display in a surprisingly small space a quite large range of topics, including correspondence between Lie groups and Lie algebras, elements of algebraic geometry and of algebraic group theory over fields of real and complex numbers, basic facts of the theory of semisimple Lie groups (real and complex; their local and global classification included) and their representa­ tions, and Levi-Malcev theorems for Lie groups and algebraic groups. VIII The Translator's Preface There is nothing comparable to this book by the broadness of scope in the literature on the group theory or Lie algebra theory. At the same time, the book is self-contained indeed since only the very basics of algebra, calculus and smooth manifold theory are really needed to understand it. It is this feature that makes it compare favourably with the books mentioned above. On the other hand, as far as algebraic groups are concerned, it cannot replace treatises like those by Humphrey or Jantzen, especially over fields of prime characteristic. Nevertheless, this book might serve better for the first acquaintance with these topics. The algebraic groups, however, though vital in the approach adopted, are not the main characters of the book while the theory of Lie groups is. Still, the viewpoint of algebraic group theory enabled the authors to simplify some proofs of important theorems. Other novelties include: - M alcev closure which enabled the authors to give a new proof of existence of an embedded (here: virtual) Lie subgroup with given tangent algebra; - a proof of the fixed point theorem for compact groups of affine transforma­ tions that does not refer to integration over the group and corollaries of this theorem, such as Weyl's theorem on unitarity of a compact linear group and the algebraicity of compact linear groups; - a generalization of V. Kac's classification of periodic automorphisms based on ideas different from those put forward by Kac originally; - a simple proof of E. Cartan's theorem on the conjugacy of the maximal compact subgroups, that does not require any Riemannian geometry. Lastly, I believe that some further reading on the rapidly developing gener­ alization of the topic of the book should be recommended, including: (Twisted) loop algebras and, more generally, Kac-Moody algebras: V.
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