Lie Groups and Algebraic Groups

346 Pages·1990·24.14 MB·English

by  Arkadij L. Onishchik, Ernest B. Vinberg (auth.)

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 1s t 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. Kac: Infinite-dimensional Lie algebras, 2nd ed. Cambridge Univ. Press,
Cambridge, 1983;
V. Kac., A. Raina: Bombay lectures on highest weight representations of
infinite dimensional Lie algebras, Adv. Series in Math. Phys. 2, World Sci.,
Singapore, 1987;
A. Pressley, G. Segal: Loop groups. Clarendon Press. Oxford, 1986.
Lie superalgebras and stringy Lie (super) algebras:
V. Kac: Lie superalgebras. Adv. Math. 26, 1977,8-96.
M. Scheunert: The theory of Lie superalgebras. An introduction. LN in Math.
# 716, Springer, Berlin, 1979.
D. Leites (ed): Seminar on supermanifolds, vols. 1 and 3. Kluwer, Dordrecht,
1990.
Finally, I wish to contribute one more problem to this compendium of prob
lems. The importance of Table 5 has been rapidly increasing oflate, in particular
with the introduction of new ideas and problems in theoretical physics by
V. Drinfeld ("quantum groups", quadratic algebras, etc.). The time and ingenuity
consuming task of acquiring similar data should be solved once and for all:
The Translator's Preface IX
Problem. Write a program for a computer to calculate data similar to those
of Table 5, e.g. S"(R(A» ® Am(R(M», etc., together with an explicit expression
for the highest (lowest) weight vectors of the irreducible components ofthe tensor
product in terms of the vectors from the initial spaces-factors 1.
I am sure that the reader will enjoy the book and treasure it as does everybody
I know, who was lucky enough to get a copy in Russian.
Petrozavodsk - Moscow-Stockholm, 1979-89 Dimitry Leites
1 As far as I know, some partial results in this area were obtained recently in Montreal and at Moscow
University.
Table of Contents
Commonly Used Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Chapter 1. Lie Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§ 1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1°. Lie Groups. . . . . . . . . . . . . . .. . . . . . . . . . .. . . . .. . . . . . . . . . . . . . 1
2°. Lie Subgroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3°. Homomorphisms, Linear Representations and
Actions of Lie Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4°. Operations on Linear Representations. . . . . . . . . . . . . . . . . . . . . . 5
5°. Orbits and Stabilizers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6°. The Image and the Kernel of a Homomorphism. . . . . . . . . . . . . . 8
7°. Coset Manifolds and Quotient Groups. .. . . . . . . . . . . . . . . . . . . 9
8°. Theorems on Transitive Actions and Epimorphisms . . . . .. . . . . 11
9°. Homogeneous Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
10°. Inverse Image of a Lie Subgroup with Respect
to a Homomorphism..... ............. ...... .... ........ 14
11 Semidirect Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
0.
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Hints to Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
§ 2. Tangent Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1°. Definition ofthe Tangent Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . 19
2°. Tangent Homomorphism....... ...... ................. ... 21
3°. The Tangent Algebra of a Stabilizer. . . . . . . . . . . . . . . . . . . . . . . . 22
4°. The Adjoint Representation and the Jacobi Identity. . . . . . . . . . 23
5°. Differential Equations for Paths on a Lie Group. . . . . . . . . . . . . 25
6°. Uniqueness Theorem for Lie Group Homomorphisms. . . . . . . . 27
7°. Exponential Map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8°. Existence Theorem for Lie Group Homomorphisms. . . . . . . . . . 30
9°. Virtual Lie Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
10°. Automorphisms and Derivations. . . . . . . . . . . . . . . . . . . . . . . . . . 35
11 The Tangent Algebra of a Semidirect Product of Lie Groups. . . 36
0.
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Hints to Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
XII Table of Contents
§ 3. Connectedness and Simple Connectedness. . . . . . . . . . . . . . . . . . . . . . . 42
1°. Connectedness........................................... 43
2°. Covering Homomorphisms........... . .... .. ...... ........ 44
3°. Simply Connected Covering Lie Groups. . . . . . . . . . . . . . . . . . . . . 45
4°. Exact Homotopy Sequence. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 46
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Hints to Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
§ 4. The Derived Algebra and the Radical. . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1°. The Commutator Group and the Derived Algebra. . . . . . . .. . . . . 50
2°. Malcev Closures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3°. Existence of Virtual Lie Subgroups ......................... 52
4°. Solvable Lie Groups. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 53
5°. Lie's Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6°. The Radical. Semisimple Lie Groups. .. . . . .. . . . . . . . . . . . . . . . . 55
7°. Complexification... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Hints to Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Chapter 2. Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
§ 1. Affine Algebraic Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1°. Embedded Affine Varieties. ...... ......................... 59
2°. Morphisms............................................. 61
3°. Zariski Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4°. The Direct Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 64
5°. Homomorphism Extension Theorems............ .......... 65
6°. The Image of a Dominant Morphism. . . . . . . . . . . . . . . . . . . . . . . 66
7°. Hilbert's Nullstellensatz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8°. Rational Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9°. Rational Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
10°. Factorization of a Morphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Hints to Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
§2. Projective and Quasiprojective Varieties. . . . . . . . .. . . . . . . . . . . . . . . 74
1°. Graded Algebras..... .. ...... .............. ....... ....... 74
2°. Embedded Projective Algebraic Varieties. . . . . . . . . . . . . . . . . . . . . 75
3°. Sheaves of Functions. . . . .. .. . . . . . . . . . . .. . . . . . .. . . . . .. . . . . 77
4°. Sheaves of Algebras of Rational Functions. . . . . . . . . . . . . . . . . . . 79
5°. Quasiprojective Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6°. The Direct Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7°. Flag Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Hints to Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85