Lie Groups and Lie Algebras I: Foundations of Lie Theory; Lie Transformation Groups [1 ed.] 9783540612223, 9783642579998

Table of contents :
I. FOUNDATIONS OF LIE THEORY
Introduction
Chapter 1. Basic Notions
1. Lie Groups, Subgroups and Homomorphisms
1.1 Definition of a Lie Group
1.2 Lie Subgroups
1.3 Homomorphisms of Lie Groups
1.4 Linear Representations of Lie Groups
1.5 Local Lie Groups
2. Actions of Lie Groups
2.1 Definition of an Action
2.2 Orbits and Stabilizers
2.3 Images and Kernels of Homomorphisms
2.4 Orbits of Compact Lie Groups
3. Coset Manifolds and Quotients of Lie Groups
3.1 Coset Manifolds
3.2 Lie Quotient Groups
3.3 The Transitive Action Theorem and the Epimorphism Theorem
3.4 The Pre-image of a Lie Group Under a Homomorphism
3.5 Semidirect Products of Lie Groups
4. Connectedness and Simply-connectedness of Lie Groups
4.1 Connected Components of a Lie Group
4.2 Investigation of Connectedness of the Classical Lie Groups
4.3 Covering Homomorphisms
4.4 The Universal Covering Lie Group
Chapter 2. The Relation Between Lie Groups and Lie Algebras
1. The Lie Functor
1.1 The Tangent Algebra of a Lie Group
1.2 Vector Fields on a Lie Group
1.3 The Differential of a Homomorphism of Lie Groups
1.4 The Differential of an Action of a Lie Group
1.5 The Tangent Algebra of a Stabilizer
1.6 The Adjoint Representation
2. Integration of Homomorphisms of Lie Algebras
2.1 The Differential Equation of a Path in a Lie Group
2.2 The Uniqueness Theorem
2.3 Virtual Lie Subgroups
2.4 The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra
2.5 Deformations of Paths in Lie Groups
2.6 The Existence Theorem
2.7 Abelian Lie Groups
3. The Exponential Map
3.1 One-Parameter Subgroups
3.2 Definition and Basic Properties of the Exponential Map
3.3 The Differential of the Exponential Map
3.4 The Exponential Map in the Full Linear Group
3.5 Cartan's Theorem
3.6 The Subgroup of Fixed Points of an Automorphism of a Lie Group
4. Automorphisms and Derivations
4.1 The Group of Automorphisms
4.2 The Algebra of Derivations
4.3 The Tangent Algebra of a Semi-Direct Product of Lie Groups
5. The Commutator Subgroup and the Radical
5.1 The Commutator Subgroup
5.2 The Maltsev Closure
5.3 The Structure of Virtual Lie Subgroups
5.4 Mutual Commutator Subgroups
5.5 Solvable Lie Groups
5.6 The Radical
5.7 Nilpotent Lie Groups
Chapter 3. The Universal Enveloping Algebra
1. The Simplest Properties of Universal Enveloping Algebras
1.1 Definition and Construction
1.2 The Poincare-Birkhoff-Witt Theorem
1.3 Symmetrization
1.4 The Center of the Universal Enveloping Algebra
1.5 The Skew-Field of Fractions of the Universal Enveloping Algebra
2. Bialgebras Associated with Lie Algebras and Lie Groups
2.1 Bialgebras
2.2 Right Invariant Differential Operators on a Lie Group
2.3 Bialgebras Associated with a Lie Group
3. The Campbell-Hausdorff Formula
3.1 Free Lie Algebras
3.2 The Campbell-Hausdorff Series
3.3 Convergence of the Campbell-Hausdorff Series
Chapter 4. Generalizations of Lie Groups
1. Lie Groups over Complete Valued Fields
1.1 Valued Fields
1.2 Basic Definitions and Examples
1.3 Actions of Lie Groups
1.4 Standard Lie Groups over a Non-archimedean Field
1.5 Tangent Algebras of Lie Groups
2. Formal Groups
2.1 Definition and Simplest Properties
2.2 The Tangent Algebra of a Formal Group
2.3 The Bialgebra Associated with a Formal Group
3. Infinite-Dimensional Lie Groups
3.1 Banach Lie Groups
3.2 The Correspondence Between Banach Lie Groups and Banach Lie Algebras
3.3 Actions of Banach Lie Groups on Finite-Dimensional Manifolds
3.4 Lie-Frechet Groups
3.5 ILB- and ILH-Lie Groups
4. Lie Groups and Topological Groups
4.1 Continuous Homomorphisms of Lie Groups
4.2 Hilbert's 5-th Problem
5. Analytic Loops
5.1 Basic Definitions and Examples
5.2 The Tangent Algebra of an Analytic Loop
5.3 The Tangent Algebra of a Diassociative Loop
5.4 The Tangent Algebra of a Bol Loop
References
II. LIE TRANSFORMATION GROUPS
Introduction
Chapter 1. Lie Group Actions on Manifolds
1. Introductory Concepts
1.1 Basic Definitions
1.2 Some Examples and Special Cases
1.3 Local Actions
1.4 Orbits and Stabilizers
1.5 Representation in the Space of Functions
2. Infinitesimal Study of Actions
2.1 Flows and Vector Fields
2.2 Infinitesimal Description of Actions and Morphisms
2.3 Existence Theorems
2.4 Groups of Automorphisms of Certain Geometric Structures
3. Fibre Bundles
3.1 Fibre Bundles with a Structure Group
3.2 Examples of Fibre Bundles
3.3 G-bundles
3.4 Induced Bundles and the Classification Theorem
Chapter 2. Transitive Actions
1. Group Models
1.1 Definitions and Examples
1.2 Basic Problems
1.3 The Group of Automorphisms
1.4 Primitive Actions
2. Some Facts Concerning Topology of Homogeneous Spaces
2.1 Covering Spaces
2.2 Real Cohomology of Lie Groups
2.3 Subgroups with Maximal Exponent in Simple Lie Groups
2.4 Some Homotopy Invariants of Homogeneous Spaces
3. Homogeneous Bundles
3.1 Invariant Sections and Classification of Homogeneous Bundles
3.2 Homogeneous Vector Bundles. The Frobenius Duality
3.3 The Linear Isotropy Representation and Invariant Vector Fields
3.4 Invariant A-structures
3.5 Invariant Integration
3.6 Karpelevich-Mostow Bundles
4. Inclusions Among Transitive Actions
4.1 Reductions of Transitive Actions and Factorization of Groups
4.2 The Natural Enlargement of an Action
4.3 Some Inclusions Among Transitive Actions on Spheres
4.4 Factorizations of Lie Groups and Lie Algebras
4.5 Factorizations of Compact Lie Groups
4.6 Compact Enlargements of Transitive Actions of Simple Lie Groups
4.7 Groups of Isometries of Riemannian Homogeneous Spaces of Simple Compact Lie Groups
4.8 Groups of Automorphisms of Simply Connected Homogeneous Compact Complex Manifolds
Chapter 3. Actions of Compact Lie Groups
1. The General Theory of Compact Lie Transformation Groups
1.1 Proper Actions
1.2 Existence of Slices
1.3 Two Fiberings of an Equi-orbital G-space
1.4 Principal Orbits
1.5 Orbit Structure
1.6 Linearization of Actions
1.7 Lifting of Actions
2. Invariants and Almost-Invariants
2.1 Applications of Invariant Integration
2.2 Finiteness Theorems for Invariants
2.3 Finiteness Theorems for Almost Invariants
3. Applications to Homogeneous Spaces of Reductive Groups
3.1 Complexification of Homogeneous Spaces
3.2 Factorization of Reductive Algebraic Groups and Lie Algebras
Chapter 4. Homogeneous Spaces of Nilpotent and Solvable Groups
1. Nilmanifolds
1.1 Examples of Nilmanifolds
1.2 Topology of Arbitrary Nilmanifolds
1.3 Structure of Compact Nilmanifolds
1.4 Compact Nilmanifolds as Towers of Principal Bundles with Fibre T1
2. Solvmanifolds
2.1 Examples of Solvmanifolds
2.2 Solvmanifolds and Vector Bundles
2.3 Compact Solvmanifolds (The Structure Theorem)
2.4 The Fundamental Group of a Solvmanifold
2.5 The Tangent Bundle of a Compact Solvmanifold
2.6 Transitive Actions of Lie Groups on Compact Solvmanifolds
2.7 The Case of Discrete Stabilizers
2.8 Homogeneous Spaces of Solvable Lie Groups of Type (I)
2.9 Complex Compact Solvmanifolds
Chapter 5. Compact Homogeneous Spaces
1. Uniform Subgroups
1.1 Algebraic Uniform Subgroups
1.2 Tits Bundles
1.3 Uniform Subgroups of Semi-simple Lie Groups
1.4 Connected Uniform Subgroups
1.5 Reductions of Transitive Actions of Reductive Lie Groups
2. Transitive Actions on Compact Homogeneous Spaces with Finite Fundamental Groups
2.1 Three Lemmas on Transitive Actions
2.2 Radical Enlargements
2.3 A Sufficient Condition for the Radical to be Abelian
2.4 Passage from Compact Groups to Non-Compact Semi-simple Groups
2.5 Compact Homogeneous Spaces of Rank 1
2.6 Transitive Actions of Non-Compact Lie Groups on Spheres
2.7 Existence of Maximal and Largest Enlargements
3. The Natural Bundle
3.1 Orbits of the Action of a Maximal Compact Subgroup
3.2 Construction of the Natural Bundle and Its Properties
3.3 Some Examples of Natural Bundles
3.4 On the Uniqueness of the Natural Bundle
3.5 The Case of Low Dimension of Fibre and Basis
4. The Structure Bundle
4.1 Regular Transitive Actions of Lie Groups
4.2 The Structure of the Base of the Natural Bundle
4.3 Some Examples of Structure Bundles
5. The Fundamental Group
5.1 On the Concept of Commensurability of Groups
5.2 Embedding of the Fundamental Group in a Lie Group
5.3 Solvable and Semi-simple Components
5.4 Cohomological Dimension
5.5 The Euler Characteristic
5.6 The Number of Ends
6. Some Classes of Compact Homogeneous Spaces
6.1 Three Components of a Compact Homogeneous Space and the Case when Two of them Are Trivial
6.2 The Case of One Trivial Component
7. Aspherical Compact Homogeneous Spaces
7.1 Group Models of Aspherical Compact Homogeneous Spaces
7.2 On the Fundamental Group
8. Semi-simple Compact Homogeneous Spaces
8.1 Transitivity of a Semi-simple Subgroup
8.2 The Fundamental Group
8.3 On the Fibre of the Natural Bundle
9. Solvable Compact Homogeneous Spaces
9.1 Properties of the Natural Bundle
9.2 Elementary Solvable Homogeneous Spaces
10. Compact Homogeneous Spaces with Discrete Stabilizers
Chapter 6. Actions of Lie Groups on Low-dimensional Manifolds
1. Classification of Local Actions
1.1 Notes on Local Actions
1.2 Classification of Local Actions of Lie Groups on ]R1, ￂﾫ:1
1.3 Classification of Local Actions of Lie Groups on ]R2 and ￂﾫ:2
2. Homogeneous Spaces of Dimension ??3
2.1 One-dimensional Homogeneous Spaces
2.2 Two-dimensional Homogeneous Spaces (Homogeneous Surfaces)
2.3 Three-dimensional Manifolds
3. Compact Homogeneous Manifolds of Low Dimension
3.1 On Four-dimensional Compact Homogeneous Manifolds
3.2 Compact Homogeneous Manifolds of Dimension ??6
3.3 On Compact Homogeneous Manifolds of Dimension ??7
References

Citation preview

Encyclopaedia of Mathematical Sciences Volume 20

Editor-in-Chief: R. V. Gamkrelidze

A. L. Onishchik (Ed.)

Lie Groups and Lie Algebras 1 Foundations ofLie Theory Lie Transformation Groups

Springer-V erlag Berlin Heidelberg GmbH

Consulting Editors of the Series:

.A. A. Agrachev, A. A. Gonchar, E. F. Mishchenko, N. M. Ostianu, V. P. Sakharova, A. B. Zhishchenko

Title of the Russian edition: Itogi nauki i tekhniki, Sovremennye problemy matematiki, Fundamental'nye napravleniya, VoI. 20, Gruppy Li i Algebry Li 1 Publisher VINITI, Moscow 1988

Mathematics Subject Classification (1991): 17Bxx, 22-XX, 22Exx, 53C30, 53C35, 57Sxx, 57Txx ISBN 978-3-540-61222-3

Library of Congress Cataloging-in-Publication Data Gruppy Li i algebry Li 1. English. Lie groups and Lie algebras II A. L. Onishchik, ed. p. cm. - (Encyclopaedia of mathematical sciences; v. 20) Translation of original Russian, issued as v. 20 ofthe serial: Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamental 'nye napravieniiil. Includes bibliographical references and index. Contents: Foundations ofLie theory I A. L. Onishchik, E. B. Vinberg - Lie groups oftransformations/V. V. Gorbatsevich, E.B. Vinberg. ISBN 978-3-540-61222-3 ISBN 978-3-642-57999-8 (eBook) DOI 10.1007/978-3-642-57999-8

1. Lie groups. 2. Lie algebras. 1. Onishchik, A.L. II. Onishchik, A.L. Foundations ofLie theory.I993. III. Gorbatsevich, V. V. Lie groups of transformations. 1993. IV. Title. V. Series. QA387.G7813 1993 512'.55-dc20 This work is subject to copyright. AII rights are reserved, whether the whole or part ofthe material is concemed, 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 ofthis publication or parts thereofis permitted only under the provisions ofthe German Copyright Law ofSeptember 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Originally published by Springer-Verlag Berlin Heidelberg New York in 1993 Softcover reprint ofthe hardcover Ist edition 1993

41/3140-543210- Printed on acid-free paper

List of Editors, Authors and Translators Editor-in-Chiif

R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul. Vavilova 42,117966 Moscow, Institute for Scientific Information (VINITI), ul. Usievicha 20 a, 125219 Moscow, Russia Consulting Editor

A L. Onishchik, Yaroslavl University, Sovetskaya ul. 14, 150000 Yaroslavl, Russia Authors

v. V. Gorbatsevich, Moscow Institute of Aviation Technology, 27 Petrovka Str.,

103767 Moscow, Russia AL.Onishchik, Yaroslavl University, Sovetskaya ul. 14,150000 Yaroslavl, Russia E. B. Vinberg, Chair of Algebra, Moscow University, 119899 Moscow, Russia Translator

A Kozlowski, Toyama International University, Toyama, Japan

Contents I. Foundations of Lie Theory A. L. Onishchik, E. B. Vinberg 1 II. Lie Transformation Groups V. V. Gorbatsevich, A. L. Onishchik

95 Author Index 231 Subject Index 232

I. Foundations of Lie Theory A. L. Onishchik, E. B. Vinberg Translated from the Russian by A. Kozlowski

Contents Introduction

4

Chapter 1. Basic Notions

6

§1. Lie 1.1 1.2 1.3 1.4

Groups, Subgroups and Homomorphisms Definition of a Lie Group Lie Subgroups Homomorphisms of Lie Groups Linear Representations of Lie Groups 1.5 Local Lie Groups §2. Actions of Lie Groups 2.1 Definition of an Action 2.2 Orbits and Stabilizers 2.3 Images and Kernels of Homomorphisms 2.4 Orbits of Compact Lie Groups §3. Coset Manifolds and Quotients of Lie Groups 3.1 Coset Manifolds 3.2 Lie Quotient Groups 3.3 The Transitive Action Theorem and the Epimorphism Theorem 3.4 The Pre-image of a Lie Group Under a Homomorphism 3.5 Semidirect Products of Lie Groups §4. Connectedness and Simply-connectedness of Lie Groups 4.1 Connected Components of a Lie Group 4.2 Investigation of Connectedness of the Classical Lie Groups 4.3 Covering Homomorphisms 4.4 The Universal Covering Lie Group

6 6 7 9 9 11 12 12 12 14 14 15 15 17 18 18 19 21 21 22 24 26

2

A. L. Onishchik, E. B. Vinberg

4.5 Investigation of Simply-connectedness of the Classical Lie Groups . . . . . . . . . . .

27

Chapter 2. The Relation Between Lie Groups and Lie Algebras

29

§1. The Lie Functor . . . . . . . . . . . . 1.1 The Tangent Algebra of a Lie Group 1.2 Vector Fields on a Lie Group . . . . 1.3 The Differential of a Homomorphism of Lie Groups 1.4 The Differential of an Action of a Lie Group 1.5 The Tangent Algebra of a Stabilizer . . . . 1.6 The Adjoint Representation . . . . . . . . §2. Integration of Homomorphisms of Lie Algebras 2.1 The Differential Equation of a Path in a Lie Group 2.2 The Uniqueness Theorem . . . . . . . . . . . . . 2.3 Virtual Lie Subgroups . . . . . . . . . . . . . . 2.4 The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra 2.5 Deformations of Paths in Lie Groups 2.6 The Existence Theorem 2.7 Abelian Lie Groups . . . . §3. The Exponential Map 3.1 One-Parameter Subgroups 3.2 Definition and Basic Properties of the Exponential Map 3.3 The Differential of the Exponential Map 3.4 The Exponential Map in the Full Linear Group 3.5 Cartan's Theorem . . . . . . . . . . . . . . 3.6 The Subgroup of Fixed Points of an Automorphism of a Lie Group . . . . . . . §4. Automorphisms and Derivations 4.1 The Group of Automorphisms 4.2 The Algebra of Derivations . 4.3 The Tangent Algebra of a Semi-Direct Product of Lie Groups §5. The Commutator Subgroup and the Radical 5.1 The Commutator Subgroup . . . . . . 5.2 The Maltsev Closure . . . . . . . . . 5.3 The Structure of Virtual Lie Subgroups 5.4 Mutual Commutator Subgroups 5.5 Solvable Lie Groups 5.6 The Radical 5.7 Nilpotent Lie Groups

29 29 31 32 34 35 35 37 37 38 38

48 48 48 50 51 52 52 53 54 55 56 57 58

Chapter 3. The Universal Enveloping Algebra

59

§1. The Simplest Properties of Universal Enveloping Algebras 1.1 Definition and Construction ............

59 60

39 40 41 43 44 44 44 46 47 47

1. Foundations of Lie Theory

1.2 1.3 1.4 1.5

The Poincare-Birkhoff-Witt Theorem Symmetrization The Center of the Universal Enveloping Algebra The Skew-Field of Fractions of the Universal Enveloping Algebra §2. Bialgebras Associated with Lie Algebras and Lie Groups 2.1 Bialgebras 2.2 Right Invariant Differential Operators on a Lie Group 2~3 Bialgebras Associated with a Lie Group §3. The Campbell-Hausdorff Formula 3.1 Free Lie Algebras 3.2 The Campbell-Hausdorff Series 3.3 Convergence of the Campbell-Hausdorff Series

3

61 63 64 64 66 66 67 68 70 70 71

73

Chapter 4. Generalizations of Lie Groups

74

§1. Lie Groups over Complete Valued Fields 1.1 Valued Fields 1.2 Basic Definitions and Examples 1.3 Actions of Lie Groups 1.4 Standard Lie Groups over a Non-archimedean Field 1.5 Tangent Algebras of Lie Groups §2. Formal Groups 2.1 Definition and Simplest Properties 2.2 The Tangent Algebra of a Formal Group 2.3 The Bialgebra Associated with a Formal Group §3. Infinite-Dimensional Lie Groups 3.1 Banach Lie Groups 3.2 The Correspondence Between Banach Lie Groups and Banach Lie Algebras 3.3 Actions of Banach Lie Groups on Finite-Dimensional Manifolds 3.4 Lie-Frechet Groups 3.5 ILB- and ILH-Lie Groups §4. Lie Groups and Topological Groups 4.1 Continuous Homomorphisms of Lie Groups 4.2 Hilbert's 5-th Problem §5. Analytic Loops 5.1 Basic Definitions and Examples 5.2 The Tangent Algebra of an Analytic Loop 5.3 The Tangent Algebra of a Diassociative Loop 5.4 The Tangent Algebra of a Bol Loop

74 74 75 75 76 76 78 78 79 80 80 81 82 83 84 85 86 87 87 88 88 89 90 91

References

92

4

A. L. Onishchik, E. B. Vinberg

Introduction The theory of Lie groups, to which this volume is devoted, is one of the classical well established chapters of mathematics. There is a whole series of monographs devoted to it (see, for example, Pontryagin 1984, Postnikov 1982, Bourbaki 1947, Chevalley 1946, Helgason 1962, Sagle and Walde 1973, Serre 1965, Warner 1983). This theory made its first appearance at the end of the last century in the works of S. Lie, whose aim was to apply algebraic methods to the theory of differential equations and to geometry. During the past one hundred years the concepts and methods of the theory of Lie groups entered into many areas of mathematics and theoretical physics and became inseparable from them. The first three chapters of the present work contain a systematic exposition of the foundations of the theory of Lie groups. We have tried to give here brief proofs of most of the more important theorems. Certain more complex theorems, not used in the text, are stated without proof. Chapter 4 is of a special character: it contains a survey of certain contemporary generalizations of Lie groups. The authors deliberately have not touched upon structural questions of the theory of Lie groups and algebras, in particular, the theory of semi-simple Lie groups. To these questions will be devoted a separate study in one of the future volumes of this series. In this entire work Lie groups, as a rule, will be denoted with capital Latin letters, and their tangent algebras with the corresponding small Gothic letters, In addition the following notation will be used:

GO - connected component of the identity of a Lie group (or a topological group) Gj G' = (G, G) - the commutator subgroup of a group Gj G(p) = (G(p-l) , GP-l))j

Rad G - the radical of a Lie group Gj rad g - the radical of a Lie algebra gj )

xy

is differentiable. In other words, the coordinates of the product of two elements have to be differentiable functions of the coordinates of the factors. With the aid of the implicit function theorem it is easy to show that in any Lie group the inverse £ :

G

-t

G, x

I--->

X-I

is also a differentiable map. Lie groups over C are called complex Lie groups and Lie groups over lR - real Lie groups. Any complex Lie group can be viewed as a real Lie group of twice the dimension. One can also consider analytic groups by requiring that the manifold G and the map I-L be analytic over the field K. Clearly, every complex Lie group is analytic, but even in the real case it turns out that in any Lie group there exists an atlas with analytic transition functions, in which the map I-L is expressed in terms of analytic functions (see 3.3 of Chap. 3).

1. Foundations of Lie Theory

7

Examples. 1. The additive group of the field K (we will denote it also by

K).

2. The multiplicative group K X of the field K. 3. 'The circle' 'll' = {z E ex : Izl = I} is a real Lie group. 4. The group GLn(K) of invertible matrices of order n over the field K, with the differentiable structure of an open subset of the vector space Ln(K) of all matrices, i.e. (global) coordinates are given by the matrix entries. 5. The group GL(V) of invertible linear transformations of an n-dimensional vector space over the field K can be regarded as a Lie group in view of the isomorphism GL(V) ~ GLn(K), which assigns to each linear transformation its matrix with respect to some fixed basis. 6. The group GA(S) of (invertible) affine transformations of an n-dimensional affine space S over the field K possesses also a canonical differentiable structure, which turns it into a Lie group. Namely, with respect to the affine coordinate system of the space S affine transformations can be written in the form X 1---+ AX + B, where X is a column vector of coordinates of a point, A an invertible square matrix and B a column vector. The entries of the matrix A and the column vector B can be taken as (global) coordinates in the group GA(S). 7. Any finite or countable group equipped with the discrete topology and the structure of a O-dimensional differentiable manifold. The direct product of Lie groups is the direct product of the corresponding abstract groups endowed with the differentiable structure of the direct product of differentiable manifolds. The Lie group Kn (the direct product of n copies of the additive group of the field K) is called the n-dimensional vector Lie group. The Lie group ~ (the direct product of n copies of the group 'll') is called the n-dimensional torus. 1.2. Lie Subgroups. A subgroup H of a Lie group G is said to be a Lie subgroup if it is a submanifold of the underlying manifold of G. Let us recall that by am-dimensional submanifold of an n-dimensional manifold X we mean a subset Y c X such that for each of its points y one of the following equivalent conditions is satisfied: (1) in a local coordinate system in some neighbourhood U of the point y the subset Y n U can be described parametrically in the form (i=l, ... ,n)

where CP1,' .. , CPn are differentiable functions defined in some domain of the space Km and the rank of the matrix ~~ at all points of this domain is equal to m. (2) in a local coordinate system in some neighbourhood U of the point y the set Y n U can be given by equations of the form

t :. . . :t::?

(i=l, ... ,n-m),

A. L. Onishchik, E. B. Vinberg

8

where 8(/1, ...

/1, ... ,fn-m are differentiable functions and the rank of the matrix ,In-Tn) at all points of the neighbourhood U is n - m.

8("1, ... '''n)

(3) in a suitable local coordinate system in some neighbourhood U of the point y the subset Y n U is given by equations Xm+l

= ... = Xn = o.

(Sometimes the terms 'submanifold' and correspondingly 'Lie subgroup' are understood in a wider sense. In this book this wider meaning is referred to by the term 'virtual Lie subgroup'; see 2.3 of Chap. 2. Lie subgroups in our sense are also known as 'closed Lie subgroups'.) Every m-dimensional submanifold of a differentiable manifold carries the structure of a m-dimensional differentiable manifold, as local coordinates on which we can take, for example, the parameters tl, ... ,tm from condition (1). Every Lie subgroup, endowed with this differentiable structure is itself a Lie group. From the topological and the differential geometric viewpoints every subgroup H of a Lie group G looks at any point h E H the same as at the identity, since it is transformed into itself by a translation (left or right) by h, which is a diffeomorphism of the manifold G. Therefore in order to verify that a subgroup H is a Lie subgroup it suffices to establish that it is a submanifold in some neighbourhood of the identity. Examples. 1. Any subspace of a vector space is a Lie subgroup of the corresponding Lie group. 2. The group l' (see Example 3 of 1.1) is a Lie subgroup of the group ex, viewed as a real Lie group. 3. Any discrete subgroup is a Lie subgroup. 4. The group of non-singular diagonal matrices is a Lie subgroup of the Lie group GLn(K). 5. The group of non-singular triangular matrices is a Lie subgroup of the Lie group GLn(K). 6. The group SLn(K) of unimodular matrices is a co dimension 1 Lie subgroup of the Lie group GLn(K). 7. The group On(K) of orthogonal matrices is a Lie subgroup of dimension n(n2-1) of the Lie group GLn(K). 8. The group SPn(K) (n even) of symplectic matrices is a Lie subgroup of dimension n(n2+1) of the Lie group GLn(K). 9. The group Un of unitary matrices is a real Lie subgroup of dimension n 2 of the Lie group GLn(C). A Lie subgroup of the Lie group GLn(V) (and in particular of GLn(K) = GL(Kn» is called a linear Lie group. As any submanifold, a Lie subgroup is an open subset of its closure. However, any open subgroup of a topological group is at the same time closed, since it is the complement of the union of its own cosets, which, like the

1. Foundations of Lie Theory

9

subgroup itself, are open subsets. Hence any Lie subgroup is closed. For real Lie groups the converse is also valid, see Theorem 3.6 of Chap. 2. 1.3. Homomorphisms of Lie Groups. Let G and H be Lie groups. A map ---+ H is a homomorphism if it is simultaneously a homomorphism of abstract groups and a differentiable map. A homomorphism I : G ---+ H is called an isomorphism if there exists an inverse 1-1 : H ---+ G, i.e. if I is simultaneously an isomorphism of abstract groups and a diffeomorphism of manifolds (however, in connection with this, see the corollary to Theorem 3.4).

I :G

Examples. 1. The exponential map x f-t eX is a homomorphism from the additive Lie group K to the Lie group K x 2. The map A f-t det A is a homomorphism from the Lie group GLn(K) to the Lie group K X 3. For any element 9 of a Lie group G the inner automorphism a(g) : x f-t gxg- 1 is a Lie group automorphism. 4. The map x f-t eix is a homomorphism from the Lie group lR to the Lie group T. 5. The map assigning to each affine transformation of an affine space its differential (linear part) is a homomorphism from the Lie group GA(S) (see Example 6 of 1.1) to the Lie group GL(V), where V is the vector space associated with S. 6. Any homomorphism from a finite or a countable group to a Lie group is a homomorphism in the sense of the theory of Lie groups. Obviously the composition of homomorphisms of Lie groups is also a homomorphism of Lie groups. 1.4. Linear Representations of Lie Groups. A homomorphism from a Lie group G to the Lie group GL(V) is called its linear representation in the space V. For example, if to each matrix A E GLn(K) we assign the transformations Ad(A) and Sq(A) of the space Ln(K), defined by the formulas

Ad(A)X = AXA-l, Sq(A)X = AXAT

,

(1)

then we obtain linear representations Ad and Sq of the Lie group GLn(K) in the space Ln (K). Sometimes one considers complex linear representations of real Lie groups or real linear representations of complex Lie groups. In the former case, it is understood that the group of linear transformations of a complex vector space is being considered as a real Lie group, in the latter - that the given complex Lie group is being considered as a real one. Let Rand S be linear representations of some group G in spaces V and U respectively. Recall that, by the sum of representations Rand S, is meant the linear representation R + S of the group G in the space V E9 U, defined by the formula

10

A. 1. Onishchik, E. B. Vinberg

(R

+ S)(g)(v + u) =

R(g)v

+ S(g)u

(2)

by the product of the representations Rand S the linear representation RS of the group G in the space V ® U, defined on decomposable elements by the formula (3) (RS) (g)(u ® v) = R(g)v ® S(g)u The sum and product of an arbitrary number of representations are defined analogously. By the dual representation of a representation R we mean the representation R* of the group G in the space V* - the dual of V, given by the formula (R*(g)f)(v) = f(R(g)-lV) (4) It is easy to see that, if Rand S are linear representations of a Lie group G, then the representations R + S, RS and R* are also linear representations of it as a Lie group (i.e. they are differentiable). For any integers k, l 2 0 the identity linear representation I d of the group GL(V) in the space V generates its linear representation Tk,l = Idk(Id*)1 in the space V ® ... ® V ® V* ® ... ® V* of tensors of type (k, l) on V. We ~

k

'-...-'" I

will give convenient interpretations of representations Tk,l in the two most commonly met cases: k = 0 and k = 1. Tensors of type (0, l) can be viewed as l-linear forms on V. For any such form f we have

(5) Tensors of type (1, I) can be viewed as I-linear maps V x ... x V any such map F we have

~

V. For

(6) The representations Ad and Sq of the group GLn(K) considered above, are just its representations in the spaces of tensors (on Kn) of type (1,1) and (2,0) respectively, expressed in the matrix form. If R is a linear representation of some group G in a space V and U c V is an invariant subspace, there is a natural way to define the subrepresentation Ru : G ~ GL(U) and the quotient representation R v / u : G ~ GL(VjU). Clearly, every subrepresentation and every quotient representation of a linear representation of a Lie group G are linear representations of it as a Lie group. A special role in group theory is played by one-dimensional representations, which are precisely the homomorphisms from the given group to the multiplicative group of the base field. They are referred to as characters 1 of the group G. Characters form a group with respect to the operation of multiplication of representations; the inverse of an element in this group is its dual representation. We will denote the group of characters of a group G by 1

Here the word character is being used in its narrower sense. In its wider sense character refers to the trace of any (not necessarily one-dimensional) linear representation.

1. Foundations of Lie Theory

11

X(G). Traditionally additive notation is used to denote its group operation, thus by definition (Xl

+ X2)(g)

= Xl (g)x2(g) (Xl,X2 E X(G)).

In the context of the theory of Lie groups characters are assumed to be differentiable. 1.5. Local Lie Groups. In certain situations it turns out to be useful to have a local version of the concept of a Lie group. By a local Lie group we mean a differentiable manifold U together with a base point e, its neighbourhood V and a differentiable map (multiplication)

JL : V x V

----t

U, (x, y)

r-t

xy

satisfying the conditions ex = xe = x and (xy)z = x(yz) for X,y,Z,XY,yz E V. These conditions imply the existence of a neighbourhood of the identity We V and a differentiable map (inversion) t:

W

----t

W,

X r-t X-I

such that xx- l = x-Ix = e for wE W. Every Lie group G can be viewed as a local Lie group by taking V = U = G. Replacing U and V by neighbourhoods of the identity Ul and VI C V n Ul , satisfying the condition VI VI C Ul , one obtains also a local Lie group, called a restriction of the original one. By transitivity restriction generates a certain equivalence relation of local Lie groups. Strictly speaking, by a local Lie group one understands an equivalence class defined in this way. Two local Lie groups are said to be isomorphic, if for some of their restrictions (Ub eb Vb JLl) and (U2, e2, V2, JL2) there is a diffeomorphism f : Ul ----t U2 satisfying the conditions f(ed = e2, f(Vl ) = V2 and f(xy) = f(x)f(y) for x, y E VI. One can easily see that isomorphism of local Lie groups is an equivalence relation. The concepts of Lie subgroup, homomorphism of Lie groups, etc. have natural local analogues and many theorems from the theory of Lie groups can be formulated for local Lie groups (some of them even turn out to be simpler). However the theory oflocal Lie groups does not have an independent status for the reason that every local Lie group a posteriori turns out to be a restriction of some Lie group. (This is a corollary of the theorem on theexistence of a Lie group with a given tangent algebra: see Theorem 2.11 of Chap. 2). Within the theory of Lie groups the significance of the concept of a local Lie group lies basically in that it enables us to use local terminology. For example, two Lie groups are said to be locally isomorphic if they are isomorphic as local Lie groups. This definition is a precise interpretation of the intuitive notion that two given Lie groups "look the same in a neighbourhood of the identity" .

12

A. L. Onishchik, E. B. Vinberg

§2. Actions of Lie Groups 2.1. Definition of an Action. A homomorphism a from a Lie group G to the group Diff X of diffeomorphisms of a differentiable manifold X is called its action on X if the map G x X ----) X, (g, x) f---t a(g)x is differentiable. Examples. 1. For any Lie group G one can define the following three actions l, r, a on itself: l(g)x = gx, r(g)x = xg-l, a(g)x = gxg- 1

2. The natural action of the group GLn(K) on the projective space p(Kn) is a Lie group action. 3. Every linear representation R : G ----) GL(V) of a Lie group G can be viewed as its action on the space V. This kind of action is called linear. 4. Analogously, every homomorphism f : G ----) GA(S) can be viewed as an action of the Lie group G on the affine space S. Such an action is called affine. Clearly, the composition of an action f : G ----) Diff X and a homomorphism f : H ----) G is an action of the Lie group H on the manifold X. In cases where there is no danger of confusion we will write simply gx in place of a(g)x. Actions of Lie groups will be considered in detail in the second part of this volume. We will use without any additional explanations certain common terms which are defined there. 2.2. Orbits and Stabilizers. Suppose we are given an action a of a Lie group G on a manifold X and let x be a point of this manifold. Consider the map ax : G ----) X, 9 f---t a(g )x. Its image is precisely the orbit a( G)x of the point x, and the pre-image of the point x is its stabilizer G x = {g E G: a(g)x = x}

The pre-images of the other points of the orbit are the cosets of G x . From the definition of a Lie group action it follows that the map ax is differentiable, and from the commutativity of the diagram ax

G -----; X I(g)

1

1

a(g)

ax

G -----; X

for any 9 E G, that it has constant rank. It is known (see, for example, Dieudonne 1960), that a differentiable map f : X ----) Y of constant rank k is linearizable in a neighbourhood of any point of the manifold X. From this it follows that: 1) the pre-image of any point y = f(x) is a submanifold of co dimension k in X, with TxU-1(y)) = Kerdxf;

13

1. Foundations of Lie Theory

2) for each point x E X there is some neighbourhood U such that its image is a submanifold of dimension k in Y, with Tf(",)U(U)) = d",f(T",(X)); 3) if f(X) is a submanifold in Y, then dimf(X) = k Proof. The last part is proved in the following way: if we had dim f(X) > k, then in view of (2) the manifold f(X) would be covered by a countable 0 number of submanifolds of lower dimension, which is impossible.

Applying this to the map a", constructed above we obtain the following theorem: Theorem 2.1. Let a be an action of a Lie group G on a differentiable manifold X. For any point x E X the map a", has a constant rank and if this constant rank is k, then: 1) the stabilizer G", is a Lie subgroup of codimension k in G and Te (G "') = Kerdea",; 2) for some neighbourhood U of the identity in the group G the set a(U)x is a submanifold of dimension k in X, and T",(a(U)x) = dea",(Te(G)); 3) if the orbit a( G)x is a submanifold in X, then dim a( G)x = k.

We remark that the orbit is not always a submanifold. (A counter-example will be given below). Assertion 1) of the theorem can be used to prove that a given subgroup H of a Lie group G is a Lie subgroup. For this purpose it suffices to realize H as the stabilizer of some point for a certain action of the Lie group G. Moreover, if the orbit of the point turns out to be a manifold of known dimension, then assertion 3) makes it possible to compute the dimension of the subgroup H. Applying these considerations to the representations Tk,l of the group GL(V) in tensor spaces (see 1.4) we find, in particular, that the group of non-singular linear transformations, preserving some given tensor, is a linear Lie group. Examples. 1. By considering the representation of the group GL(V) in the space B+(V) of symmetric bilinear forms (symmetric tensors of type (0,2)) we see that the group O(V, f) of non-singular linear transformations preserving a given symmetric bilinear form f is a linear Lie group. If the form f is non-degenerate, then its orbit is open in B+(V) and, therefore,

dimO(V,f) = dimGL(V) - dimB+(V) = n(n 2-1) where n = dim V. 2. Analogously, by considering the representation of the group GL(V) in the space B_(V) of alternating bilinear forms, we see that the group Sp(V, f) of non-singular linear transformations preserving a given alternating bilinear form f is a linear Lie group. If the form f is non-degenerate, then dim Sp(V, f)

= dim GL(V) -

dim B_ (V)

= n(n 2+ 1)

14

A. L. Onishchik, E. B. Vinberg

3. By considering the representation of the group GL(V) in the space of bilinear operations on V (tensors of type (1,2)) we see that the group of automorphisms of any algebra is a linear Lie group. 2.3. Images and Kernels of Homomorphisms. Let f : G - H be a homomorphism of Lie groups. Define an action a of G on the manifold H by the formula a(g)h = f(g)h, where the right hand side is the product of elements in H. In other words, a is the composite of the homomorphism f and the action l of H on itself by left translations. Let e be the identity of the group H. Then a e = f, a(G)e = f(G) and the stabilizer of the point e under the action a is just the kernel Ker f of the homomorphism f. Applying Theorem 2.1 to the action a and the point e E H, we obtain the following theorem Theorem 2.2. Let f : G - H be a homomorphism of Lie groups. Then f is a map of constant rank and if this rank is equal to k, then 1) Ker f is a Lie subgroup of codimension kin G, and Te(Ker f) = Kerdef· 2) For some neighbourhood U of the identity in the group G the set f(U) is a submanifold of dimension k in Hand Te(f(U» = def(Te(G)). 3) if f(G) is a Lie subgroup of H, then dimf(G) = k. Example. Consider the homomorphism det : GLn(K) - KX. Its kernel is the group SLn(K) of unimodular matrices. Since det GLn(K) = K X we have rkdet = 1 and hence SLn(K) is a Lie subgroup of co dimension 1 in GLn(K). Clearly, if f(G) is a submanifold, then f(G) is a Lie subgroup in H. The following example shows that f(G) is not always a submanifold. Let f : IR.1m be a homomorphism given by the formula f(x)

=

(e ia1 "', .•. ,eian "')

(ab'" ,an E IR.)

It is known (see, for example, Bourbaki 1947), that if the numbers ab'" ,an are linearly independent over Q, then the set f(lR.) is dense in 1m (this is the so called dense winding of the torus), and therefore, for n > 1 is not a submanifold. In order that the set f(lR.) be a submanifold it is necessary and sufficient for the numbers al, ... ,an to be commensurable. 2.4. Orbits of Compact Lie Groups. The preceding example makes the following assertion particularly interesting. Theorem 2.3. Every orbit of an action of a compact Lie group is a submanifold. Proof. Let a be an action of a compact Lie group G on a manifold X and let x E X. We will prove that the orbit a(G)x is a submanifold in X. For this purpose it is enough to verify that it is a submanifold in a neighbourhood of

I. Foundations of Lie Theory

15

the point x. Let U be a neighbourhood of the identity in the group G such that a(U)x is a submanifold of X. The orbit a(G)x is a union of nonintersecting sets a(U)x and a(C)x, where C = G\UG",. Since the set UG", is open in G its complement C is closed and thus compact; but then the set a(C)x = a",(C) is also compact and therefore closed in X. Thus the intersection of the orbit a( G)x with the open set X \ a( C)x containing the point x, is a submanifold. D Corollary. A homomorphic image of a compact Lie group is a Lie subgroup.

The most important examples of compact Lie groups (besides the finite ones) are the n-dimensional torus 'II""', the orthogonal group On (= On (IR)) and the unitary group Un. In order to prove the compactness of the group On we note that it is the subset of the space Ln (IR) of all real matrices determined by the algebraic equations Ek aikajk = bij, and hence is closed in Ln(IR). The same equations imply the inequalities laij I :::; 1 which show that the group On is bounded in Ln{lR). Analogously one proves the compactness of the group Un·

§3. Coset Manifolds and Quotients of Lie Groups 3.1. Coset Manifolds. On the set of cosets of a Lie subroup of a Lie group one can naturally introduce the structure of a differentiable manifold. In giving an axiomatic description of this structure we will make use of several definitions. Let X and Y be differentiable manifolds and let p be a differentiable map from X onto Y. For any function f defined on an open submanifold U C Y we define a function p* f on p -1 U by the formula

(p* f)(x)

= f(p(x))

The map p is called a quotient map if 1) A subset U C Y is open if and only if p-1(U) is open in X. 2) A function f defined on an open subset U c Y is differentiable if and only if the function p* f is differentiable. The map p is called a trivial fibre bundle with fibre Z (where Z is also a differentiable manifold) if there is a diffeomorphism v: Y x Z ~ X satisfying the condition p(v(y, z)) = y. The map p is called a locally trivial fibre bundle with fibre Z if the manifold Y can be covered by subsets, such that the map p is a trivial fibre bundle with fibre Z over each of them. Every locally trivial fibre bundle is a quotient map. (It suffices to check this for trivial fibre bundles.) As an example of a quotient map, which is not a fibre bundle, one can take the map z --+ Z2 of the complex plane onto itself. A quotient map possesses the following universality property: given a commutative triangle

16

A. L. Onishchik, E. B. Vinberg

in which p is a quotient map and q a differentiable map, the map ¢ is differentiable. If q is also a quotient map and ¢ is bijective, then ¢ is a diffeomorphism. The latter fact can be interpreted in the following way: if we are given a map p from a differentiable manifold X to a set Y, then Y posesses at most one differentiable structure for which p is a quotient map. Theorem 3.1. Let H be a Lie subgroup of a Lie group G. The set G / H of left cosets of H in G possesses a unique differentiable structure for which the canonical map p: G -+ G/H, 9 t--t gH is a quotient map. In addition 1) the map p is a locally trivial fibre bundle 2) the canonical action of the group G on G / H (by left translations) is differentiable. Proof. We introduce a topology on the set G / H so that a subset U c G / H is open if and only if p-1(U) is open in G. It is easy to see that this makes the map p continuous and open, and the space G / H Hausdorff. Let now 8 c G be a submanifold transversal to H at the point e. Consider the map v: 8 x H -+ G, (s, h) t--t sh

Since v(s,e) = sand v(e,h) = h d(e,e) v(ds, dh)

= ds + dh

so that d(e,e)v is an isomorphism of the tangent spaces. Hence, there exist neighbourhoods 8 1 and V of the point e in 8 and H respectively, such that v maps 8 1 x V diffeomorphic ally onto an open subset of the group G. Since v(s,hh') = v(s,h)h', the map v is a local diffeomorphism everywhere on 8 1 xH. Let 8 2 be a neighbourhood of the point e in 8 1 such that 8 2 -1 8 2 nH c V. Then v is injective on 8 2 x H. Thus by initially choosing a suitable cross section 8 we can assume that v maps 8 x H diffeomorphically onto an open subset of the group G. Under the map p the cross section 8 maps bijectively onto some neighbourhood U of the point p( e) in the space G / H. We transfer by means of this map the differentiable structure of 8 onto U. The map p becomes then a trivial fibre bundle over U. Furthermore, for any 9 E G we transfer the differentiabie structure from U to gU via the left translation leg). Since the map pis equivariant with respect to left translations, this definition of a differentiable structure on gU turns the map p into a trivial fibre bundle over gU and hence into a quotient map. For gl, g2 E G the differentiable stuctures, defined on gl U and g2 U coincide on the intersection gl Un g2 U, since over it the map p is a quotient map with

17

1. Foundations of Lie Theory

respect to any of these structures. Thus we defined a differentiable structure on G I H with respect to which the map p is a locally trivial fibre bundle (and consequently a quotient map). The natural action of the group G on G I H is defined by the map A: G x GIH

-t

GIH,

(g',gH)

f---+

g'gH

which fits into a commutative diagram

GxG idxp

G

1

G x GIH

where JL is the multiplication in the group G. The map id x p is a locally trivial fibre bundle and therefore a quotient map. From the universality of quotient maps it follows that A is a differentiable map. 0 From assertion 1) of the theorem it follows that the map

is surjective and has Te(H) as its kernel. Therefore the space Tp(e) (G I H) can be canonically identified with Te (G) ITe (H). 3.2. Lie Quotient Groups Theorem 3.2. Let N be a normal Lie subgroup of a Lie group G. Then the quotient group GIN with the differentiable structure defined in Theorem 3.1 is a Lie group. Proof. The differentiability of the multiplication JL N on GIN is proved analogously to the proof of Part 2) of Theorem 3.1, with the help of the commutative diagram

GxG

GIN x GIN

-----> I-'N

GIN

o

Under the canonical bijection between the subgroups of a group G containing N and subgroups of the group GIN, Lie subgroups in G correspond to Lie subgroups in GIN and conversely. If the group G acts on a differentiable manifold X so that the action of N is trivial, then the induced action on X of the quotient group GIN is differentiable.

18

A. L. Onishchik, E. B. Vinberg

3.3. The Transitive Action Theorem and the Epimorphism Theorem Theorem 3.3. Let a be a transitive action of a Lie group G on a differentiable manifold X. Then for any x E X the map

{3,,: GIG"

---t

X,

gG"

I-t

a(g)x

is a diffeomorphism, which commutes with the action of G. (It is assumed that the group G acts on GIG" by left translations. ) Proof. We have a commutative triangle G

X GIG"

where a,,(g) = gx. Since p is a quotient map the map (3" is differentiable. According to Theorem 2.1 rka" = dimX = dim GIG" so that dea,,(Te(G)) = T,,(X) and dp(e){3" is an isomorphism of tangent spaces. Thus (3" is a diffeomorphism. 0 N

Theorem 3.4. Let f : G = Ker f. Then the map

---t

H be an epimorphism of Lie groups and let

¢: GIN

---t

H,

gN

I-t

f(g)

is an isomorphism of Lie groups. Proof. It suffices to apply Theorem 3.3 to the action a of the group G on H constructed in 2.3. 0 Corollary. A bijective homomorphism of Lie groups is an isomorphism. 3.4. The Pre-image of a Lie Group Under a Homomorphism Theorem 3.5. Let f : G ---t H be a homomorphism of Lie groups and let HI be a Lie subgroup in H. Then G I = f-I(Hd is a Lie subgroup in G and

Te(Gd = (de!)-l(Te(HI)). Proof. Consider the composite a = {3 0 f of the homomorphism f and the canonical action (3 of the Lie group H on HI HI. The subgroup G I is then the stabilizer of the point pee) E HIHI (where p is the canonical projection of H onto HIH I ) under the action a. By Theorem 2.1 it is a Lie subgroup and Te(G I ) = Kerdeap(e) = Ker (deP 0 de!).

Since KerdeP

= Te(HI ), we have

19

1. Foundations of Lie Theory

D Corollary 1. Let HI, H2 be Lie subgroups of a Lie group G. Then HI n H2 is also a Lie subgroup and Te(HI n H 2)

= Te(Hd n Te(H2)'

Proof. This is proved by applying the theorem to the inclusion HI C G and the subgroup H2 C G. D

Note that the intersection of submanifolds is not, in general, a submanifold. Corollary 1 can be trivially extended to any finite number of subgroups. It also holds for an infinite family of subgroups (see Theorem 4.2).

Corollary 2. Let R : G ----; GL(V) be a linear representation of a Lie group G and let U C V be any subspace. Then G(U) = {g E G : R(g)U C U}

is a Lie subgroup of G and Te(G(U)) = {~ E Te(G) : (dR)(OU C U}. Proof. The proof consists of applying the theorem to the homomorphism R and the subgroup GL(V; U)

=

{A E GL(V) : AU C U}

of the group GL(V), which is an open subset of the space

D

L(V; U) = {X E L(V) : XU C U}.

Corollary 3. Under the assumptions of Corollary 2 let W C V be any subspace contained in U. Then G(U, W) = {g E G : (R(g) - E)U

c

W}

is a Lie subgroup of G and Te(G(U, W))

=

{~ E

Te(G) : (dR)(~)U C W}.

Proof. The proof consists of applying the theorem to the homomorphism

R and the subgroup

GL(V; U, W) = {A E GL(V) : (A - E)U

c

W}

of the group GL(V), which is an open subset of the space L(V; U, W)

= {X

E L(V) : XU C W}.

D

3.5. Semidirect Products of Lie Groups. In many cases it is most convenient to describe the structure of a Lie group by means of the concept of semi-direct product.

20

A. L. Onishchik, E. B. Vinberg

Let us recall that by the semi-direct product of abstract groups G 1 and G 2 we mean the direct product of the sets G 1 and G 2 equipped with a group operation by means of the formula

(7) where b is some homomorphism from the group G 2 to the group Aut G 1 of automorphisms of the group G 1 . We will denote the semi-direct product by G 1 ) O} so 7rl(SLn(lR)) = 7r(GL~(lR)). Finally, consider the action of the group SPn(lR) on IRn {O}) = 7r2(lRn \ {O}) = 0 for n ~ 4 so

\

{O}. As 7rl(lRn \

o

Chapter 2 The Relation Between Lie Groups and Lie Algebras The basic method of the theory of Lie groups, which makes it possible to obtain deep results with striking simplicity, consists in reducing questions concerning Lie groups to certain problems of linear algebra. This is done by assigning to every Lie group G its "tangent algebra" g, which to a large extent determines the group G, and to every homomorphism f : G - t H of Lie groups a homomorphism df : 9 - t ~ of their tangent algebras, which to a large extent determines the homomorphism f. In the language of category theory we have a functor from the category of Lie groups into the category of Lie algebras, whose properties are very close to those of an equivalence of categories. In honour of the founder of the theory of Lie groups we will call this functor (following M. M. Postnikov (1982)) the Lie functor.

§1. The Lie Functor 1.1. The Tangent Algebra of a Lie Group. The most direct method of defining the tangent algebra of a Lie group G consists in the following. Choose a coordinate system in a neighbourhood of the identity e of the group G so that the point e is the origin. The column vector of coordinates of a point x will be denoted by x. Consider the Taylor series of the coordinates of the product xy. Since ey = y and xe = x we have

(1) where

Q

is a bilinear vector-valued form.

30

A. L. Onishchik, E. B. Vinberg

Switching the order of x and y we obtain

(2) We see that non-commutativity of the multiplication in the group G is reflected only in terms of degree 2: 2. The group commutator (x, y) = xyx-1y-l serves as a measure of non-commutativity. Terms of degree two in the Taylor expansion of the commutator (x, y) can easily be found from the relation (x, y)yx = xy. Comparing (1) and (2) we obtain

(3) where

'}'(x,1)) = a(x,1)) - a(y, x).

(4)

Let us now define on the tangent space Te(G) a bilinear "commutator" (~, 11) 1--+ [~, 11] by the formula

(5) where "( denotes the column of coordinates of a tangent vector ( in the coordinate system of the space Te (G) associated with the chosen local coordinate system on G. The above definition can be given a coordinate free form. Let g(t) and h( s) be differentiable paths on G such that

g(O) = h(O) = e, Then [~,11]

{j2

g'(O) = ~,

h'(O) = 11.

= 8tas (g(t), h(s))lt=s=o.

(6) (7)

(The right hand side of this equality has an invariant meaning since differentiation with respect to t gives, for any s, an element of the tangent space Te (G) and the subsequent differentiation with respect to s is a differentiation of a path in Te(G).) The space Te (G) with the operation [ , ] defined in this way, is the tangent algebra of the group G and will be denoted by g. In general, the tangent algebra of a Lie group denoted by any upper case Latin letter is denoted by the corresponding lower case Gothic letter. It is clear from the definition that the tangent algebra is anticommutative, i.e. it satisfies the identity

(8) The tangent algebra of an abelian Lie group is an algebra with trivial multiplication. Let A be a (finite-dimensional) associative algebra with an identity e and G = A x its multiplicative group of invertible elements. The group G has a natural differentiable structure as an open subset of the vector space A.

I. Foundations of Lie Theory

31

Moreover, it is a Lie group and its tangent space at any point can be naturally identified with A. The identity (e

+ a)(e + b)

= e

+ a + b + ab

shows that a(a, b) = abo Therefore the commutator in the algebra 9 has the form [a, b] = ab - ba. In particular, if A = L(V) is the algebra of linear operators on the space V, then A x = GL(V). Thus the tangent algebra of the Lie group GL(V) is the space L(V) with the commutator [X,Y] = XY - YX.

(9)

It is denoted by gl(V). In the matrix case we find that the tangent algebra of the Lie group GLn(K) is the space Ln(K) of matrices with the commutator given by (9). It is denoted by gln(K). Obviously, the tangent algebra of a Lie subgroup of a Lie group G is a subalgebra of the algebra g. In particular, the commutator in the tangent algebra of any linear Lie group is given by the formula (9).

1.2. Vector Fields on a Lie Group. It is possible to give a definition of the tangent algebra of a Lie group in which the commutator arises from the commutator (Lie bracket) of vector fields. With the help of left or right translations one can construct natural isomorphisms between tangent spaces of a Lie group G at different points. Let l(g) denote left translation by 9 and r'(g) right translation by g. Then for any ~ E Th(G) let

ge = (dl(g))(e) E Tgh(G),

eg = (dr'(g))(e) E Thg(G).

From the associativity of group multiplication we derive the following identities: (gh)~ =g(h~), (g~)h =g(eh), (eg)h= ~(gh)

e

for any g,h E G, E T(G) If, in particular, G = AX is the group of invertible elements of an associative algebra A, then the "products" ge and ~g coincide with the products in the sense of the algebra A. In a local coordinate system in a neighbourhood of the identity formula (1) after differentiation with respect to the first or the second factor at the point e gives (10) ~y = ~ + a(~, 1]) + 0(11112) (e E Te(G)). X1J

= Tj + a(x,

m+ 0(lxI

2)

(1J E Te(G)).

(11)

For every ~ E Te(G) consider the right invariant vector field ~* on G given by the formula

(12)

32

A. L. Onishchik, E. B. Vinberg

Clearly, the map e f-7 e* is an isomorphism from the vector space Te (G) to the vector space T* (G) of all right invariant vector fields on the group G, with = e*(e). As the commutator of vector fields is invariant under arbitrary diffeomorphisms, the commutator of right invariant vector fields is itself right invariant. Thus T* (G) is an algebra with respect to the operation of taking the commutator of vector fields 1. This algebra can be viewed, by definition, as the tangent algebra of the group G. We shall next prove that the map ef-7 e* is an isomorphism of algebras.

e

Proof. According to (10) in a local coordinate system in a neighbourhood of the identity we have:

(13) Therefore

o It is well known that the commutator of vector fields satisfies the Jacobi identity. Therefore the commutator in the tangent algebra 9 of a Lie group G also satisfies the Jacobi identity:

[[e, 1]], (l

+ [[1], (], el + [[(, el, 1]l = 0

(14)

Every algebra with an operation [ , 1 satisfying the anti-commutativity identity (8) and the Jacobi identity (14) is called a Lie algebra. Thus the tangent algebra of any Lie group is a Lie algebra (usually called the Lie algebra of the Lie group). 1.3. The Differential of a Homomorphism of Lie Groups. Let f : G --> H be a homomorphism of Lie groups. From any definition of the tangent algebra it easily follows that the map def : 9 --> ~ is a homomorphism of Lie algebras. In cases where this cannot lead to misunderstanding we will denote it simply by df. Let N be the kernel of the homomorphism f and let neg be its tangent algebra. According to Theorem 2.2 of Chap. 1 n is the kernel of the homomorphism df and therefore an ideal of the algebra g. Any normal Lie subgroup N of a Lie group G is the kernel of the canonical homomorphism p : G --> GIN. Hence, its tangent algebra is an ideal of the algebra g. By considering the homomorphism dp we see that the tangent algebra of the quotient Lie group GIN is canonically isomorphic with the quotient algebra gin. 1

Definitions of the commutator (Lie bracket) of vector fields in different texts may differ as to sign. Here we take the following definition: [€,1J]i = L(1JjOj€i -€jOj1Ji), i.e. [€,1J] =

1J~ -

fr;o

33

I. Foundations of Lie Theory

A homomorphism from any Lie algebra 9 to the Lie algebra g[(V) is called its linear representation in the space V. The differential of a linear representation of a Lie group is a linear representation of its tangent algebra in the same space. Examples. 1. Consider the homomorphism

det : GLn(K)

----t

K

X •

Using the explicit expression for the determinant we find that

= tr X.

(dE det)(X)

Hence, the tangent algebra s[n(K) of the group SLn(K) consists of all matrices with trace zero. 2. The differentials of linear representations Ad and Sq of the group GLn(K), defined in 1.4 of Chap. 1, have the form (dAd(X))Y

E

=

XY - YX,

(dSq(X))Y

=

XY

+ YXT.

Proof. To prove, let us say, the first of these formulas, consider the path

+ tX in the group GLn(K). We have (E + tX)-l = E -

tX

+ O(t 2 ),

so that (Ad(E

+ tX))Y = (E + tX)Y(E -

tX + O(t2))

=Y

+ t(XY -

Y X)

+ O(t2). o

Let Rand S be linear representations of a Lie group G in spaces V and U respectively. Then d(R + S)(~)(v d(RS)(~)(v 0 u)

=

+ u)

(dR(~)v) 0

(dR*(~)f)(v)

+ dS(~)u, u + v 0 (dS(~)u),

= dR(~)v

=

(15)

(16) (17)

-f(dR(~)v).

Proof. Let us prove, for example, formula (16). Let g(t) be a differentiable path in the group G satisfying the conditions g(O) = e, g'(O) = ~. Then d(RS)(~)(v 0 u)

d

= dt

=

d

dt (RS)(g(t))lt=o(v 0 u)

(RS)(g(t))(v 0 u)lt=o

= (dR(Ov) 0

= u

=

d

dt (R(g(t))v 0 S(g(t))u)lt=o

+v 0

(dS(~)u).

0

With the help of these formulas one can compute the differential of the product of an arbitrary number of linear representations and their duals and, in particular, the differential Tk,l of the natural linear representation Tk,l of the group GL(V) in the space of tensors of type (k, l) (see 1.4 of Chap. 1)

34

A. L. Onishchik, E. B. Vinberg

Let us give interpretations of the representations differentiating formulas (5) and (6) of Chap. 1: (To,1

(X)f(v1,"" VI) = -

TO,I

and

L f(vI, ... , XVi"'"

T1,1,

obtained by

vd

(18)

1.4. The Differential of an Action of a Lie Group. Even though the group of diffeomorphisms is not a Lie group, it does indisputably possess the tangent algebra - the algebra of vector fields. Correspondingly, the differential of an action of a Lie group G on a manifold X ought to be a homomorphism of the algebra g into the algebra of vector fields on X. The precise definition is given below. Let a be an action of a Lie group G on a differentiable manifold X. To every element, E g we assign a vector field da(') = on X defined by the formula (20)

e

where g(t) is any differentiable path in G satisfying the conditions g(O) = e, g'(O) = ,. The field is called the velocity field of the action a corresponding to the element, E g. The map da is a homomorphism of the algebra g to the algebra of vector fields on X.

e

Proof. Let g(t) and h(s) be differentiable paths in G, satisfying conditions (6). Then by (7) -

(j2

[',77Hx) = {)t{)s (g(t), h(s»xlt=s=o. Differentiating with respect to t we obtain a vector

e(x) - dh(s)(e(h(s)-1x»

E

T",(X).

Differentiating with respect to s we obtain

[[;j](x)

= (iJe -

ei])(x)

= [e, iJ](x).

o

Examples. 1. If a = R is a linear action of a group in a vector space V, then e(v)

= dR(,)v,

where dR in the right hand side is viewed as the differential of a linear representation. 2. The group SL 2 (K) acts naturally on the projective line Kp1 = KU{ oo}:

( ac b)x=ax+b. d cx+d The algebra s[2(K) has a basis

(21)

35

I. Foundations of Lie Theory

E+

=

(~ ~),

H

=

(~ ~1)'

E_

=

(~ ~).

Differentiating (21) with respect to a, b, c, d we obtain

E+(x) = 1,

H(x) = 2x,

E_(x) = _x 2.

1.5. The Tangent Algebra of the Stabilizer. The following theorem is a reformulation of one of the assertions of Theorem 2.1 of Chap. 1. Theorem 1.1. Let Q be an action of a Lie group G on a differentiable manifold X, and let gx be the tangent algebra of the stabilizer G x of a point x E X. Then gx = {~ E 9 : dQ(~)(x) = O}. This theorem is a very effective tool for determining tangent algebras of Lie subgroups. In particular, with its help we can determine the tangent algebra of a linear Lie group defined by the requirement of preservation of some tensor. Examples. 1. The group G of non-singular linear transformations of a space V, preserving a given bilinear form f, is the stabilizer of the form f under the linear representation TO,2 of GL(V). Applying formula (18), we find that the tangent algebra of G consists of all linear transformations of V antisymmetric with respect to f. 2. In the case when V is a complex vector space, the analogous assertion holds also for any sesqui-linear form f. In particular, the tangent algebra of the group Un of unitary matrices consists of all skew-Hermitian matrices. 3. The group Aut Qt of automorphisms of a finite-dimensional algebra Qt is the stabilizer of the structure tensor of the algebra Qt under the linear representation T 1 ,2 of the group GL(Qt). Applying formula (19), we see that the tangent algebra of the group Aut Qt consists of all linear transformations D of the space Qt satisfying the condition D(ab)

= (Da) b + a (Db).

Such transformations are known as derivations of the algebra Qt. They, consequently, must form an algebra with respect to the commutator (which, of course, can also be verified directly). This algebra is denoted by DerQt. 1.6. The Adjoint Representation. Every Lie group G has a natural linear representation in its tangent algebra g. Namely, for any element 9 E G we consider the inner automorphism

a(g) : x

1-+

gxg- 1

of G. Its differential at the point e we will denote by Ad(g). It is an automorphism of the algebra g. Since a(glg2) = a(gI)a(g2)' the map Ad: G

1-+

GL(g),

9

1-+

Ad (g),

is a linear representation of G. It is referred to as the adjoint representation.

36

A. L. Onishchik, E. B. Vinberg

In the notation of 1.2 we can write (22) In particular if G = A x is the group of invertible elements of an associative algebra A, then Ad(g) is simply the conjugation by the element 9 in the algebra A. The differential of a linear representation of a Lie group G is a linear representation of its tangent algebra 9 in the space g. It is known as the adjoint representation of the algebra 9 and is denoted by ad. From formulas (10) and (11) it follows that (23) As Ad(G) C Autg, so ad(g) c Derg (see Example 3 of 1.5). On the other hand, for every algebra 9 with an operation [ , lone can define by means of formula (23) a linear map ad : 9 --) 9[(9). It is easy to see that, in the presence of anticommutativity, the Jacobi identity is equivalent to any of the following properties 1) the map ad is an algebra homomorphism; 2) ad(g) c Der 9 Thus we obtain yet two more proofs (and two interpretations) of the Jacobi identity in the tangent algebra of a Lie group. The following standard facts about centralizers and normalizers are connected with the adjoint representation. Proposition 1.2. For any element 9 E G its centralizer Z(g) is a Lie subgroup with Lie algebra

3(9) = {e E 9 : Ad(g)e =

0

(24)

(known as the centralizer of the element 9 in the algebra g). Proof. The subgroup Z(g) is just the stabilizer of the point 9 under the action a of G on itself by inner automorphisms. It is, therefore, a Lie group. By Theorem 1.1 its tangent algebra consists of E 9 such that da(e)(g) = 0; but it is easy to see that

e

o

e

Proposition 1.3. For any element E g, its centralizer Z(e) in G given by the formula (25) Z(e) = {g E G : Ad(g)e = 0,

is a normal Lie subgroup with tangent algebra

(known as the centralizer of the element Proposition 1.4. For any subspace

by

~

c

e in the algebra g).

(26)

g, its normalizer N(~) in G defined

I. Foundations of Lie Theory

N(~)

= {g E G : Ad(g)~

C ~},

37

(27)

is a Lie subgroup with tangent algebra n(~)

=

{~ E 9 : [~, ~l

c

~}

(28)

(known as the normalizer of the subspace ~ in the algebra g). Proof. The assertion is proved by applying Corollary 2 of Theorem 3.5 of Chap.! to the adjoint representation of G. 0

§2. Integration of Homomorphisms of Lie Algebras In this and the following sections we shall be considering paths in differentiable manifolds. By a path we shall be understand a differentiable map of a connected subset of the real line which is not just a single point, into the given manifold (real or complex). In the majority of cases, this subset (the domain of definition of the path) will not be explicit ely given. 2.1. The Differential Equation oCa Path in a Lie Group. For any path g(t) in a Lie group G we have, according to 1.2, g'(t) = ~(t)g(t);

(29)

where ~(t) E g. The path ~(t) in the algebra 9 is called the velocity of the path g(t). The identity (29) can be viewed as an equation determining the path g(t) in terms of its velocity ~(t). In a local coordinate system it turns into a system of ordinary differential equations of the first order with respect to the coordinates of g(t). Therefore the path g(t) is uniquely determined by its velocity ~(t) and the initial condition g(to) = go. On the other hand, the path g(t)h for any h E G also satisfies equation (29). Hence, all solutions of this equation can be obtained from one another by right translations. Let us consider now the question of existence of solutions to equation (29). Proposition 2.1. Suppose we are given a differentiable map t f--t ~(t) from a connected subset T C lR to the algebra g. Then there exists a solution of equation (29) defined for all t E T. Proof. It suffices to prove the assertion for the case when T is a segment. Further, it is enough to prove that there exists an c > 0 such that, for any to E T, there exists a solution of equation (29), defined for It - tol < c. Moreover, in view of the invariance of the set of solutions with respect to right translations one can suppose that g(to) = e. In a coordinate system in a neighbourhood of the identity of G equation (29) takes the form (30) g'(t) = F(~(t),g(t)),

A. L. Onishchik, E. B. Vinberg

38

where F is a differentiable vector valued function, depending only on the chosen local coordinate system. Assuming that the identity is the origin of the coordinate system, we shall denote by R a positive number for which the chosen coordinate neighbourhood contains the ball IXI :::; R. Let also C

= max 1~(t)l, tET

M

=

max IF(X, Y)I. IXI::;c,IYI::;R

Then by the well known theorem on the existence of solutions of systems of differential equations (see for example (Dieudonne 1960)) equation (30) has a solution defined for It - tol < ~, t E T. Since ~ does not depend on to it can be chosen as €. 0 2.2. The Uniqueness Theorem Theorem 2.2. A homomorphism I, from a connected Lie group G to a Lie group H, is uniquely determined by its differential.

Proof. Any element 9 E G can be connected with the identity by a path

g(t),O :::; t :::; 1 Let g(t) satisfy equation (29) with initial condition g(O)

= e. Applying to this equation the homomorphism I, we find that the path h(t) = I(g(t)) in the group H satisfies the equation

h'(t) = dl(e(t))h(t)

with initial condition h(O) = e. This determines the element I(g) = h(l).

0

2.3. Virtual Lie Subgroups. As we have seen in 2.3 of Chap. 1, the image of a Lie group under a homomorphism is not always a Lie subgroup. The more general subgroups obtained in this way can serve in some cases as surrogates of Lie subgroups. Let us call a subgroup H of a Lie group G, which is equipped with the structure of a Lie group in such a way that the identity inclusion i : H ~ G is a homomorphism of Lie groups, a virtual Lie subgroup. In this situation we shall consider the algebra ~ as embedded in the algebra 9 by means of the homomorphism di. Clearly, any Lie subgroup (with the induced Lie group structure) is a virtual Lie subgroup. If 1 : H ~ G is any homomorphism of Lie groups, the group I(H), equipped with the Lie group structure of the quotient HIKer I, is a virtual Lie subgroup of the group G with tangent algebra dl(~). The topology of a virtual Lie subgroup may be different from the topology induced from the ambient Lie group. This is clearly seen from the example of the dense winding of the torus 1m, which carries the structure (and, in particular, the topology) of the Lie group lR but intersects any nonempty open subset of the torus in a subset which is unbounded in lR. However, from Theorem 2.2 of Chap. 1 it follows that any virtual Lie subgroup H contains a neighbourhood V of the identity, which is a submanifold

I. Foundations of Lie Theory

39

of the ambient Lie group (and, in particular, possesses the induced topology), and moreover Te(V) = ~. Global topological structure of virtual Lie subgroups is clarified by the following Proposition 2.3. Let H be a virtual Lie subgroup of a Lie group G. There exists a neighbourhood V of the identity in H and a submanifold S C G, containing the identity, such that v : S x V --+ G, (s, h) f-+ sh is a diffeomorphism of the direct product S x V onto some neighbourhood U of the identity in G. In addition H n U = TV, where T = H n S is at most a countable set. If the neighbourhood V is connected, then it is the connected component of the identity of the intersection H n U in the induced topology.

Proof. The neighbourhood V and the submanifold S can be constucted as in the proof of Theorem 3.1 of Chap. I. The count ability of T follows from the fact that H can contain at most a countable family of mutually nonintersecting open subsets. To prove the last assertion one makes use of the fact that every countable subset of]Rn is totally disconnected. 0

The following theorem makes it possible to give a topological characterization of virtual Lie subgroups of real Lie groups. Theorem 2.4 (Yamabe 1950). Every path connected subgroup of a real Lie group is a virtual Lie subgroup. Corollary. Virtual Lie subgroups of real Lie groups coincide with subgroups having (in the induced topology) at most a countable set of path connected components. 2.4. The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra Theorem 2.5. Let G 1 and G 2 be virtual Lie subgroups of a Lie group G. If G 1 C G 2, then G 1 is a virtual Lie subgroup of the Lie group G 2 and 91 C g2. Conversely, if 91 C 92 and the group G 1 is connected, then G 1 C G 2 •

Proof. To prove the first assertion of the theorem we have to show that the identity inclusion of G 1 in G 2 is differentiable. With the help of Proposition 2.3, applied to G 2 , we see that a sufficiently small connected neighbourhood of the identity in G 1 is contained in a neighbourhood of the identity in G 2 , which is a submanifold in G. From this follows the required differentiability. To prove the second assertion consider a path g(t) in G 1 with velocity '(t) and initial condition g(t) = e. Since '(t) C 91 C 92 one can find in G 2 a path with the same velocity and the same initial condition (Proposition 2.1). Being a path in G it must coincide with g(t). Thus g(t) E G 2 • 0 Corollary 1. If virtual Lie subgroups coincide as subsets, they carry the same Lie group structure.

40

A. L. Onishchik, E. B. Vinberg

Corollary 2. A connected virtual Lie subgroup is uniquely determined by its tangent algebra (as a subalgebra of the tangent algebra of the ambient Lie group).

Not every subalgebra of the tangent algebra is the tangent algebra of some Lie subgroup. However, consideration of virtual Lie subgroups makes the picture of the correspondence between Lie subgroups and Lie subalgebras more complete. Theorem 2.6. Every subalgebra of the tangent algebra of a Lie group is the tangent algebra of some (uniquely defined) connected virtual Lie subgroup.

A proof of this theorem will be given in 5.3. Theorem 2.7. The normalizer N(H) of a connected virtual Lie subgroup H of a Lie group G is a Lie subgroup, the tangent algebra of which coincides with the normalizer n(~) of the subalgebra ~ in the algebra g. Proof. As gHg- 1 (g E G) is a connected virtual Lie subgroup with tangent algebra Ad(g)~, we have N(H) = N(~) and the assertion of the theorem follows from Theorem 1.4. 0

Corollary. A connected virtual Lie subgroup H of a connected Lie group G is normal if and only if the subalgebra ~ is an ideal of the algebra g. Theorem 2.8. The centralizer Z(H) of a connected virtual Lie subgroup H of a Lie group G is a Lie subgroup, whose tangent algebra coincides with the centralizer 3(~) of the subalgebra ~ of the algebra g. Proof. Replacing G with N(H) we can suppose that H is a normal subgroup. In virtue of Theorem 2.2, the inner automorphism a(g) of G is the identity on H if and only if its differential Ad(g) is the identity on ~. Thus, Z(H) is the kernel of the linear representation Ad~ of G. Hence, it must be a Lie subgroup and its tangent algebra coincides with the kernel of the linear representation ad~ of the algebra g, with 3(~). 0

Corollary. The centre Z (G) of a connected Lie group G is a (normal) Lie subgroup, whose tangent algebra coincides with the center 3(9) of g. (The center of a Lie algebra is the collection of elements whose commutators with all alements of the algebra are zero.)

2.5. Deformations of Paths in Lie Groups. By a deformation of a path in a differentiable manifold X we shall mean a differentiable map

T xS

---+

X,

(t, s)

f--->

x(t, s),

where T, S c lR are connected subsets, not consisting of a single point. We shall view s as the deformation parameter and the map t f---> x( t, s), for a fixed s, as the deformed path.

1. Foundations of Lie Theory

41

Proposition 2.9. Let (t, s) f--+ g(t, s) be a deformation of a path in a Lie group G. Let the elements e(t,s),,,,(t,s) E g be defined by the equations {

8g~~,8) 89~~8)

= e(t,s)g(t,s), = ",(t, s)g(t, s).

(31)

Then (32)

Proof. Differentiating in a coordinate system the first equation of (31) with respect to s and the second with respect to t and comparing the results we obtain: ae(t,s)* a",(t,s)* as +",(t,s)*e(t,s)* = at +e(t,s)*",(t,s)*,

(33)

e

where (t, s) * and ",{ t, s) * are right invariant vector fields corresponding to e{t,s) and ",{t,s) (see 1.2). Since for e,,,, E g

",*e* - e*",* (33) is equivalent to (32).

e

= [e*, ",*] = fe, ",]*, o

The elements (t, s) and ",{ t, s) have the following interpretation: for a fixed s, e{t, s) is the velocity of the path which is being deformed, and ",(t, s) is the velocity of the deformation. Equation (32) can be viewed as a differential equation with respect to t for the velocity of the deformation, which makes it possible to determine it from the velocity of the path and a given initial condition ",{to, s) = ",o{s). 2.6. The Existence Theorem Theorem 2.10. Let G and H be Lie groups with G simply connected. Then for every homomorphism ¢ : g ~ b there is a homomorphism f : G ~ H, such that df = ¢.

Proof. In order to define the image of an element 9 E G we connect it with the identity by a path g(t), 0 ~ t ~ 1, and find the velocity e{t) of this path. Further, we consider a path h{t), 0 ~ t ~ 1 in H with velocity ¢(e(t)) and initial condition h{O) = e. The element 1{g) will be taken, by definition, as h{l). Since the path g{t) is not unique we must show that the above definition does not depend on it. This is the most difficult part of the proof. Let go{t) and gl{t) be two paths in G which connect e with g. We shall denote by ho{t) and hl{t) the corresponding paths in H. We have to show that ho{l) = h1 (1). As the group G is simply connected there exists a deformation of go{t) into gl{t) i.e. a differentiable map (t, s) f--+ g{t, s) of the square Q = [0,1] x [0,1] into the group G, possessing the following properties:

42

A. L. Onishchik, E. B. Vinberg

1) g(t,O) = go(t), g(t, 1) = gl(t); 2) g(O, s) = e, g(l, s) = g. Let ~(t,s),'T}(t,s) E 9 be the elements defined by the equations (31). According to Proposition 2.9 they are connected by the relation (32). In addition, from the property 2) it follows that

'T}(O,s)

=

'T}(l,s)

= 0.

Next we define a differentiable map (t, s) f--+ h( t, s) of the square Q into H as the solution of the differential equation with respect to t

Oh~; s) = ¢(~(t, s))h(t, s) with initial condition h(O, s) = e. This is a deformation of the path ho(t) into the path hl (t). Let (( t, s) E I) be the velocity of this deformation, i.e.

oh(t, os s) = ."r( t, s )h( t, s ) According to Proposition 2.9 we have

O(~; s)

_

o¢(~:' s)) = [¢(~(t, s)), ((t, s)].

View the last equality as a differential equation with respect to t for (( t, s). Applying the homomorphism ¢ to (32), we see that this equation is satisfied by ¢('T}(t, s)). Since ((0,8) = ¢(ry(O, s)) = 0, we have ((t, s) = ¢('T}(t, s)). In particular, ((1, s) = ¢('T}(1, s)) = 0. This means that h(l, s) = const and, therefore, ho(l) = hd1). Thus we have defined a map f : G ~ H. Let us prove that f is a homomorphism. Let gl (t) and g2 (t), ~ t ~ 1 be paths in G, connecting e with gl and g2 respectively, with 6 (t) and 6 (t) as their velocities. The path connecting e with glg2 can be defined by the following equalities

°

°

g(t) _ {g2(2t), ~ t ~ ~, gl(2t-1)g2, ~~t~1. (with a suitable choice of the paths gdt) and g2(t) the map t differentiable.). Its velocity ~(t) is defined by the equalities t _ {

~( ) -

°

26 (2t), ~ t ~ ~, 26 (2t - 1), ~ ~ t ~

f--+

g(t) will be

1.

Consequently, if h l (t),h 2(t) and h(t) are paths in H corresponding to the paths gl(t), g2(t) and g(t), then

°

h(t) _ {h 2(2t), ~ t ~ ~, hl(2t-1)h2' ~~t~1.

43

I. Foundations of Lie Theory

In particular f(glg2)

=

h(l)

= h1(I)h2(1) =

f(gl)f(g2).

With the help of a change of parameter, we obtain that any path g(t) E G with velocity ~(t) and initial condition g(O) = e is taken, by the map f, to a path h(t) E H with velocity ¢(~(t)) with initial condition h(O) = e. Consequently, the map f is differentiable and def = ¢. 0

Corollary. A simply connected Lie group is determined up to an isomorphism by its tangent algebra. There is also the following theorem, various proofs of which will be given in one of the future volumes of this series (see also Postnikov 1982).

Theorem 2.11. Every finite-dimensional real (complex) Lie algebra is the tangent algebra of some real (complex) Lie group. 2.7. Abelian Lie Groups. The vector Lie group Kn is the unique simply connected Lie group whose tangent algebra is abelian 2. Hence, every connected Lie group is isomorphic to a Lie group of the form K n If, where f is a discrete subgroup of the group K n (see Theorem 4.7 of Chap. 1). If f1 and f 2 are two discrete subgroups of the group Kn, then the Lie groups K nIf 1 and and Kn If 2 are isomorphic if and only if there exists an automorphism of the Lie group Kn (i. e. some nonsingular linear transformation of the vector space Kn), which takes f1 to f 2. Every discrete subgroup of the group IRn can be taken, by a suitable automorphism, to one of the subgroups fk = {(Xl, ... ,Xk,O, ... ) E IRn

:

Xl,··· ,Xk E Z},

where k = 0, ... ,n (Bourbaki 1947). Thus we obtain the following classification of connected abelian real Lie groups.

Theorem 2.12. Every connected abelian real Lie group is isomorphic to a Lie group of the form uk x IRI. The classification of abelian complex Lie groups is considerably more complicated. For example, every connected one-dimensional complex Lie group is isomorphic to one of the Lie groups C,

C/Z

~ C*

and

A(u) = Cj(Z

+ Zu),

where u E C, Im( u) > 0, with the Lie groups A( u) and A( v) isomorphic (as complex Lie groups) if and only if

v=:::~, (~ ~)ESL2(Z). Thus connected compact one-dimensional complex Lie groups are parametrized by points of the quotient space of the complex upper half-plane by the 2

An abelian Lie algebra is an algebra with trivial multiplication.

44

A. L. Onishchik, E. B. Vinberg

action of Klein's modular group, which, as is well known, can be given in a natural way the structure of the complex plane C

§3. The Exponential Map 3.1. One-Parameter Subgroups. A path g(t) in a Lie group G, defined for all t E JR, is called a one-parameter subgroup if

g(t + s) = g(t)g(s) (and then automatically g(O) = e, g( -t) = g(t)-l). In other words, a oneparameter subgroup is a homomorphism from the Lie group JR into G. Sometimes, however, the term one-parameter subgroup is used to denote the image of such a homomorphism. A one-parameter subgroup in this sense is a virtual Lie subgroup (but need not be a genuine Lie subgroup). Proposition 3.1. A path g(t) in a Lie group G is a one-parameter subgroup if and only if its velocity ~(t) is constant and g(O) = e.

Proof. Let g(t) be a path with velocity ~(t) and initial condition g(O) = e. For any s E JR the path gs(t) = g(t + s) has velocity ~s(t) = ~(t + s) and satisfies the initial condition gs(O) = g(s). From this it follows that if ~(t) = const, then gs(t) = g(t)g(s). Conversely, if gs(t) = g(t)g(s) for all s E JR, then ~s(t) = ~(t) for all s E JR, i.e. ~(t) = const. 0 For any ~ E 9 we shall denote by g€(t) the one-parameter subgroup with velocity ~(t) == ~. We shall refer to the vector ~ as its direction vector. If G = A x is the group of invertible elements of an associative algebra A, then ga(t) = expta, where the exponential is understood as the sum of the series:

L:;. n. 00

expa=

n

(34)

n=O

(In the case when A is the matrix algebra this fact amounts to the contents of the theory of systems of linear differential equations with constant coefficients. ) One-parameter subgroups of the vector Lie group Kn are one-dimensional subspaces of the vector space Kn. More precisely, gv(t) = tv (with the usual identification of To (Kn) with Kn). 3.2. Definition and Basic Properties of the Exponential Map. For any Lie group G we set by definition exp~

= ge(1)

(~E

g).

45

1. Foundations of Lie Theory

The map exp : 9 ----t G defined in this way is known as the exponential map. In the case when G is the group of invertible elements of an associative algebra, it coincides with with the map defined by means of the series (34). In the case when G is the vector Lie group, the exponential map is the identity. By means of a linear change of the parameter t we see that

ge(t) =

(35)

expt~

The theorem about the differentiable dependence of the solutions of a system of differential equations on the parameters shows that the map exp is differentiable and from (35) it follows that its differential at zero is the identity map. From this, in turn, follows Proposition 3.2. The exponential map exp : 9 :----t G maps a certain neighbourhood of zero in the tangent algebra 9 diffeomorphically onto a neighbourhood of the identity of G.

By an analogous method we can prove a more general assertion. Proposition 3.3. Let 9

= al E9 ... E9 ak be a decomposition of a Lie algebra

9 as a direct sum of subspaces. Then the map

6 + ... + ~k

f-t

exp ~i •.• exp ~k

(~i E

ai),

maps some neighbourhood of zero in the algebra 9 diffeomorphically onto a neighbourhood of the identity in G.

These properties of the exponential map make it possible to choose certain special coordinate systems in a neighbourhood of the identity of G. Namely, let {ell ... ,en} be a basis of the algebra g. Then each of the maps (tb ... , tn)

f-t

(tl, ... , tn)

exp(tlel

f-t

+ ... + tne n ),

exphel ... exptne n ,

defines a diffeomorphism of some neighbourhood of zero in the space Kn onto a neighbourhood of the identity in G. Defined in this way coordinates in a neighbourhood of the identity in G are called canonical coordinates of the first and second kind respectively. In general, the exponential map does not posssess any good properties globally. As we shall see in the following paragraphs it need not be surjective, injective, open etc. A homomorphism f of Lie groups takes the one-parameter subgroup with direction vector ~ to the one-parameter subgroup with direction vector df(~). Consequently, f(exp~) = expdf(~).

(36)

(This means that in canonical coordinates of the first kind, every Lie group homomorphism is expressed as a linear map.) In particular Ad(exp~)

= expad(~)

(37)

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A.1. Onishchik, E. B. Vinberg

Example. Consider the homomorphismdet: GLn(K) tr (Example 1 of 1.3) so det exp X = e trX

----+

KX. Since ddet =

(X E Ln(K)).

The property of multiplicativity, which characterizes the usual exponentiation, is satisfied in the case of the exponential map in a Lie group only in a restricted sense. Proposition 3.4. If

[~,

TJ 1 = 0, then exp(~+TJ) =exp~exPTJ·

Proof. If [~, TJ 1 = 0, then there exists a homomorphism f : K2 ----+ G for which df(a, b) = a~ + bTJ (Theorem 2.10). Hence, it suffices to prove the assertion for a vector group; but in this case it is obvious. 0 In particular, if a Lie group G is abelian, the map exp is a homomorphism of the vector group 9 into G. 3.3. The Differential of the Exponential Map. In order to compute the differential of the map exp : 9 ----+ G at a point ~ E g, we consider the deformation of the path in G, defined by the formula

g(t, s) =

expt(~

+ STJ)

(38)

(so that for any S the deformed path is a one-parameter group). We have (d{exp) (TJ) = TJ(l)exp~, where TJ(t) = TJ(t,O) is the velocity of deformation (38) for s = O. As the velocity of the deformed path is e(t, s) + STJ, according to Proposition 2.9, we have,

e

TJ'(t) = [~, TJ(t) 1+ TJ

(39)

with initial condition TJ(O) = O. The solution of the equation (39) can be written in the form () _ exptad(~) -1( ) TJ t t ad (~) tTJ , where

ex p ;: ~ 1

for a linear operator A is understood as the sum of the series

expA - 1 A

=

f

An. n=o(n+1)!

In particular, for t = 1 we obtain

(d{ exp )( TJ) =

exp ad (~) - 1 ad (~) (TJ) exp ~ .

(40)

(This formula is a special case of the formula of Helgason (1964), for the differential of the exponential map in an arbitrary linear connection space.) From formula (40) it follows that the kernel of the linear map d{exp in the case K = IC is the sum of the eigenspaces of the operator ad (0 corresponding

47

1. Foundations of Lie Theory

to eigenvalues of the form 2rrik, where k E Z, k i=- 0, and in the case K = lR the real part of this sum. We shall denote the dimension of this sum by v(e). Theorem 3.5 (Nono 1960). The map exp : 9 -+ C is a local diffeomorphism at a point e E 9 if and only if the operator ad(e) has no eigenvalues of the form 2rrik, k i=- o. If this condition is not satisfied, then not only the map exp is not a local diffeomorphism but it also is not open at the point e. The set exp-l(expe) is a closed submanifold 3 of the algebra g. Its connected component containing coincides with the connected component of the set Ad( Z (exp e)) e and has dimension v( e).

e

3.4. The Exponential Map in the Full Linear Group. It is easy to see that the exponential of a Jordan block with eigenvalue A is similar to a Jordan block with eigenvalue eA. Hence, the exponential map in GLn(C) is surjective. The exponential map in GLn(lR) is not surjective, its image consisting of matrices which have an even number of Jordan blocks of every order corresponding to each negative eigenvalue. This image is neither open nor dense in GLn(lR). The exponential map in SLn(C) is not surjective. Its image is dense, but it does not contain, for instance, a Jordan block with eigenvalue different from 1 (but which is an n-th root of 1). However, in PSLn{C) = PGLn(CC) the exponential map is surjective, just as in GLn(C). The image of the exponential map in SLn{lR) is described just as in GLn(lR). In each of the groups GLn(C), GLn(lR), SLn(lR) the exponential map defines a diffeomorphism of the open subset of the tangent algebra, consisting of matrices all of whose eigenvalues A satisfy the condition IImAI < rr, onto the open subset of the group, consisting of the matrices without negative eigenvalues (Morinaga 1950). 3.5. Cartan's Theorem. One of the applications of the exponential map is the proof of the following theorem, which gives a topological characterization of Lie subgroups of real Lie groups. Theorem 3.6 (E. Cartan's Theorem). Every closed subgroup of a real Lie group is a Lie subgroup. Proof. Let H be a closed subgroup of a real Lie group C. Let us denote E 9 and by T the set of elements e E 9 for which there exist sequences C E lR such that -+ 0, -+ e and exp E H. It is easy to see that the numbers Cn can be taken to be integers. In this case we obtain:

n

en

a

cnen

expe

en

en

= lim (expen)C

n

E H.

acnen ae so that ae E T.

Moreover, for any E lR we have Let e, 'Tl E T. Consider the path

-+

h(t) = expte exp t'Tl E H. 3

This submanifold can have connected components of different dimension.

48

A. L. Onishchik, E. B. Vinberg

For sufficiently small t we have h(t)

((0) = 0,

= exp((t), where ((t)

is a path in g, with

('(0) = h'(O) = ~ + 'fl.

Consequently, lim n (( .!.) E T. n Thus T is a subspace in 9 and expT c H. Let S subspace. Consider the map

~ + 'fI =

c

9 be a complementary

¢ : 9 ---? G, According to Proposition 3.3, it gives a diffeomorphism of some neighbourhood U of zero in the algebra 9 onto a neighbourhood of the identity in the group G. We shall prove that, for a sufficiently small neighbourhood U,

H n ¢(U) = ¢(T n U) (= exp(T n U)).

(41)

Let us suppose that equality (41) does not hold for any choice of U. Then there must exist a sequence 'fin E S \ {O}, such that 'fin ---? 0 and exp 'fin E H. Passing to a subsequence we can ensure that Cn'fln ---? 'fI E S \ {O} for some Cn E JR, but in this case 'fI E T which is impossible. 0 The complex analogue of Cartan's Theorem is false, since any real Lie subgroup of a complex Lie group is closed, but is not necessarily a complex Lie subgroup.

3.6. The Subgroup of Fixed Points of an Automorphism of a Lie Group. In the special case when H is the subgroup of fixed points of some automorphism, the exponential map makes it possible not only to show that H is a Lie subgroup, but also to find its tangent algebra, and moreover this applies in equal measure to real and complex Lie groups. Theorem 3.7. Let a be an automorphism of a Lie group G. Then

G tr

= {g

E

G : a(g)

= g}

is a Lie subgroup with tangent algebra

Proof. The assertion of the theorem follows directly from the fact that the automorphism a commutes with the exponential map (formula (36)). 0

§4. Automorphisms and Derivations 4.1. The Group of Automorphisms. Let G be a Lie group, Aut G-the group of its automoprphisms (as a Lie group), Autg-the group of automorphisms of its tangent algebra.

1. Foundations of Lie Theory

49

If G is connected, then the map d : Aut G --; Aut g, which assigns to each automorphism of G its differential, is an injection (Theorem 2.2), and if G is simply connected, then it is an isomorphism (Theorem 2.10). The group Autg is a linear Lie group (Example 3 of 2.2 of Chap. 1). Using these facts one can, in the case of a simply connected group G, transfer the Lie group structure of Aut 9 to Aut G. This makes the action of the group Aut G on G differentiable. In the general case we have Proposition 4.1. For any connected Lie group G the group d Aut G is a Lie subgroup of the group Aut g. Proof. By Theorem 4.7 of Chap. 1 we have G = GIN, where G is the simply connected Lie group with the same tangent algebra and N is a discrete central subgroup of it. The group Aut G can be naturally identified with a subgroup of Aut G consisting of the automorphisms which preserve N. It contains a subgroup H consisting of the automorphisms which fix N. By Theorem 4.2 of Chap. 1 H is a Lie subgroup of Aut G (as the intersection of the stabilizers of the points of N), and from the discreteness of N it follows that in some neighbourhood of the identity of Aut G the subgroups Aut G and H coincide. D

Thus, for any connected Lie group G the group Aut G possesses a natural structure of a Lie group. The inner automorphisms of the group G form a normal subgroup in Aut G, which is isomorphic to G I Z, where Z is the center of G, and which is denoted by Int G. If G is connected, then dInt G = Ad G depends only on the algebra 9 (see 4.2) and is a normal subgroup of Aut g. It is called the group of inner automorphisms of the algebra 9 and is denoted by Int g. Being the image of the group G under the adjoint representation the group Int 9 is a virtual Lie subgroup (but not necessarily a genuine Lie subgroup) of Aut g. Correspondingly, the group Int G is a virtual Lie subgroup of Aut G. The quotient group Aut GlInt G (which can be given the structure of a Lie group when Int G is a Lie subgroup of Aut G), is referred to, somewhat loosely, as the group of outer automorphisms of the Lie group G. Analogously, the quotient group Aut glInt 9 is called the group of outer automorphisms of the algebra g. In the case of a simply connected group G there is a natural isomorphism Aut GlInt G '::: Aut glInt g. Examples. 1. Let G be a connected abelian Lie group. Then Aut 9 = GL(g), Intg = {E}, and the group dAutG consists of the automorphisms of the algebra 9 preserving the kernel of the exponential map (which can be any discrete subgroup of the vector group g). 2. Let G be the Lie group of matrices of the form

A. L. Onishchik, E. B. Vinberg

50

Xn

o

Z

o

YI

1

Yn 1

Its tangent algebra 9 has a basis {Xl,'" X n , Yb ... , Yn , Z} for which [Xi, YiJ = Z and the remaining commutators of the basis vectors are O. (This Lie algebra is known as the Heisenberg algebra.) The subspace 3 = (Z) is the centre of g. Any automorphism has to preserve it, i.e. multiply Z by a number c #- 0, and induce in g/3 a linear transformation which multiplies by c- l the skew-symmetric bilinear form f, defined by the following rule: f(X i , Yi) = 1 and is 0 on the remaining pairs of basis vectors. Conversely, any linear transformation of 9 with these properties is an automorphism. As for the inner automorphisms, they have the form

The group Int 9 is a Lie subgroup of Aut 9 and is isomorphic to the vector group K2n. The quotient group Aut glInt 9 is an extension of the group SP2n (K) by the group K*. 3. Let G be the group of affine transformations of the line. This Lie group is isomorphic to the group of matrices of the form

(~ ~)

=1=

0),

= (~ ~) , satisfying the relation [X, YJ = Y. One verifies directly that Aut 9 = Ad( G) ~ G. In the complex case the group G is connected and Intg = Ad(G) = Autg.

whose tangent algebra consists of matrices X

=

(~ ~)

(where a

and Y

In the real case G consists of two connected components (distinguished by the sign of a) and Int 9 = Ad(GO) is a subgroup of index 2 in Aut g. In both cases AutG = IntG.

4.2. The Algebra of Derivations. The tangent algebra of the group Aut 9 is the algebra Der 9 of derivations of the algebra 9 (Example 3 of 1.5). The tangent algebra of the group Int 9 is the image of the algebra 9 under the homomorphism ad = dAd. This, in particular, shows (see Corollary 2 of Theorem 2.5) that the group Intg does not depend on which G is chosen from among the connected Lie groups having 9 as their tangent algebra. Derivations of the form ad(~), ~ E 9 are called inner derivations of the algebra g. They form an ideal in the algebra Der g. More precisely,

[D, ad(~)J = for any D E Derg,

~ E

g.

ad(D~)

(42)

51

1. Foundations of Lie Theory

4.3. The Tangent Algebra of a Semi-Direct Product of Lie Groups. Corresponding to semi-direct products of Lie groups (see 3.5 of Chap. 1) we have semi-direct sums 4 of Lie algebras. By a semi-direct sum of Lie algebras 91 and 92 we mean the direct sum of the vector spaces 91 and 92, with the Lie algebra structure given by the formula

where (3 is some homomorphism of the Lie algebra 92 into the Lie algebra Der 91. We shall denote it by 91 -9 92 or more precisely by 91 -9 92· f3

Elements of the form (6,0) (respectively (0,6)) form a sub algebra of 91 -992 isomorphic to 91 (respectively 92), which we shall identify with 91 (respectively 92). The sub algebra 91 is an ideal and

(43) The sub algebra 92 is an ideal if and only if (3 sum coincides with the direct sum 91 EB 92·

= 0, in this case the semi-direct

Example. Let V be some vector space which we consider as an abelian Lie algebra. Then Der V = 9[(V). For any linear representation p : 9 - t 9[(V) of a Lie algebra 9 one can form the semi-direct sum V -9 9, in which V is an p

abelian ideal. We say that a Lie algebra 9 decomposes as a semi-direct sum of sub algebras 91 and 92 if 1) the subalgebra 91 is an ideal 2) the algebra 9, as a vector space, is a direct sum of the subspaces 91 and 92· In this case we have the isomorphism

where (3 : 92 - t Der 91 is the homomorphism given by formula (43). In this situation we shall write 9 = 91 -992 or 9 = 92 Et 91

Proposition 4.2. The tangent algebra of a semi-direct product G 1 ) 2. By its definition (see 0 1.1. Chap. 2) the tangent algebra coincides with g. We remark that the methods described here still do not allow us to prove the existence of a global Lie group with a given tangent algebra. We shall denote by CH(g) the local Lie group with multiplication (1) constructed from a Lie algebra g in Theorem 3.4. One can verify that its one Hence the parameter subgroup with tangent vector e Egis the line t 1---+ exponential map exp : g - t Te(CH(g)) = g is the identity map.

te.

Corollary 1. For any Lie group G the operation of multiplication written in canonical coordinates of the first type has the form (11). In particular any Lie group is analytic. Proof. Let g be the tangent algebra of G. By Theorem 2.10 of Chap. 2 there exists an isomorphism of local Lie groups h : CH(g) - t G such that deh = id. Since the exponential mapping for CH(g) is the identity, it follows from formula (36) of Chap. 2 that H coincides with the exponential mapping exp for the group G. Therefore exp(e * 'T}) = (expe)(exp'T}) for all sufficiently 0 small e,'T} E G.

It is not hard to verify that the correspondence g 1---+ CH(g) defines a functor from the category of finite-dimensional Lie algebras over K (= lR or

74

A. L. Onishchik, E. B. Vinberg

O ILI>O

where CML E K and the summation is performed over all ordered tuples M,L chosen out of n non-negative integers. There exists a unique set H such that

= (Hih::;i::;n, where Hi

F(x,H(x))

E

K[[Xl, ... , xnll,

= F(H(x),x) = OJ

we have

Hi(X) = -Xi

+

L

CMX M ,

IMI>o

where CM E K. Terms of degree 2 in formula (1) define a certain bilinear map b : K n x K n - t Kn. Putting

[x,y] = b(x,y) - b(y,x), we obtain a bilinear operation in K n , which, as it turns out, makes K n into a Lie algebra. This algebra is called the "tangent algebra" of the formal group. For example, in the case where the formal group F = FG is associated with a Lie group G (see 1.1 of Chap. 2) the tangent algebra of the formal group CH(g) (see for Example 5 in 2.1) coincides with g. It is not hard to verify that the differential at 0 (Le. the linear part) of a homomorphism of formal groups is a homomorphism of their tangent algebras. Thus we obtain a functor from the category of formal groups over a ring K into the category of Lie algebras over K (the Lie functor). If K is a field of characteristic 0, then this functor and the functor 9 - t CH(g) define an equivalence of these categories (see also below 2.3). In particular, a formal group over a field of characteristic 0 is determined up to an isomorphism by its tangent algebra. In the case of a field of prime characteristic this is not true, as is shown by Example. Let K be a field of characteristic p > o. Then the one-dimensional additive and multiplicative formal groups over K (see Examples 1 and 2 of 2.1) are not isomorphic. At the same time they have the same tangent algebra - the one-dimensional abelian Lie algebra over K. For other examples see (Manin 1963), (Lazard 1955). (Manin 1963) is devoted to the classification of abelian formal groups over fields of prime characteristic. The theory of formal groups is also studied in the following books: (Dieudonne 1973) and (HazewinkeI1978).

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A. L. Onishchik, E. B. Vinberg

2.3. The Bialgebra Associated with a Formal Group. Analogously to the case of Lie groups (see §2 of Chap. 3), with every formal group one can associate some bialgebra. Let us denote by U the K-submodule in (K[[XI' .. . ,xn]])* consisting of linear forms equal to 0 on some power of the ideal m C K[[xl, ... , x n ]], generated by the elements Xl, ... , x n . The module U is free and possesses a basis (~M), dual to the system of monomials (xM):

~ M (xL) =

{Io

for M for M

=L

i= L.

Moreover, U* = K[[XI, ... , x n ]]. Operations in U arise from consideration of duality. As in 2.3 of Chap. 3, it turns out that the multiplication in the algebra K[[XI, .. . ,xnll defines a comultiplcation D in U. Furthermore, substitution into a power series of the power series Fi defines a map K[[XI, . .. ,xnll ~ K[[XI, ... , Xn , Yl, ... , Yn]], extending the associative multiplication in U. It turns out that U is a bialgebra with respect to these operations. Let 9 = Kn be the tangent algebra of our formal group. We define a map T : 9 ~ U by the formula

L ai~ei, n

T(al, ... ,an ) =

i=l

where ei = (0, ... ,1, ... ,0). One can verify that T is a homomorphism of the algebra 9 into the Lie algebra L(U). Hence T can be extended to an algebra homomorphism U(g) ~ U which we shall also denote by T. Theorem 2.2. If K is an algebra over Q (for example a field of characteristic 0), then the homomorphism T : U(g) ~ U is an isomorphism of bialgebras. Corollary. If K is an algebra over Q than the Lie functor and the functor 9 1-+ CH(g) define an equivalence of the category of formal groups over K and the category of Lie algebras over K which are free K -modules of finite rank.

§3. Infinite-Dimensional Lie Groups The most immediate infinite-dimensional generalizations of Lie groups are Banach (in particular Hilbert) Lie groups, to which case almost all basic statements of the finite-dimensional theory can be transferred. However, Banach Lie groups have a rather restricted domain of applicability and, in particular, very rarely appear as groups of transformations of finite-dimensional manifolds (for example, there are no known examples of transitive and effective actions of infinite-dimensional Banach Lie groups on compact manifolds). Significantly larger (although more difficult to study) is the class of Lie groups modelled on Fnkhet spaces, to which, in particular, belongs the group of all diffeomorphisms of an arbitrary compact COO-manifold. Research into groups

I. Foundations of Lie Theory

81

of diffeomorphisms has also lead to the definition ofILB- (or ILH- ) Lie groups, i.e. topological groups which can be represented as an inverse limit of Banach (or Hilbert) Lie groups (see Adams 1985, Omori 1974). In connection with infinite-dimensional Lie groups and their applications in physics see (Adams 1985) and also the article (Milnor J., Remarks on infinite-dimensional Lie groups. Relativite, groupes et topol. II. Les Houches, Ec. d'ete phys. theor. Sess. 40, 1983. Amsterdam e. a., 1984, 1007-1058). In this survey we do not touch upon the theory of Lie pseudo-groups of transformations (i.e. "infinite transformation groups" in the sense of Lie and Cartan) and the related theory of infinite-dimensional filtered Lie algebras. 3.1. Banach Lie Groups (see Bourbaki 1971, Hamilton 1982). Let K be a complete valued field. The definition of a Banach (Hilbert) Lie group G over K is a word for word restatement of the definition of a usual Lie group, with only this difference that G has to possess the structure of a Banach (respectively Hilbert) manifold (for the details see Bourbaki 1971, where the theory of Lie groups is presented at exactly this level of generality). Any ordinary Lie group over K is, of course, a Banach (and Hilbert) Lie group. We shall now give some infinite-dimensional examples. Examples. 1. Let M and N be real manifolds of class Coo, with M compact. Then for any k ~ 0 the set Ck(M, N) of all maps M ---+ N of class C k can be given the structure of a real Banach COO-manifold. In order to describe this structure we choose any Coo Riemannian structure on N and denote by Expy the corresponding exponential mapping at the point yEN. Let

O}, on which GLn(lR) acts by the formula

gx = Idetglx

(g E GLn(lR),x E

~).

A section of a fibre bundle p : E---+ B is a differentiable mapping s : B ---+ E such that po s = id. We denote by r(E) the set of all sections of a bundle E. Note that sections of the trivial bundle E = B x F can be identified with differentiable mappings B ---+ F. Further, a principal bundle possesses sections if and only if it is equivalent to a trivial bundle. For a vector bundle the set of sections is a vector space over the corresponding field. Sections of the bundle T(M) are vector fields on M, sections of the bundle of positive definite symmetric bilinear forms are Riemannian structures on M, sections of the bundle of positive densities are positive densities on M. Note the following simple criterion for the existence of a section: if the fibre F is homeomorphic to lRn , then r(E) -=I- 0 (see Steenrod 1951).

118

V. V. Gorbatsevich, A. L. Onishchik

From this it easily follows, for example, that on any differentiable manifold there exist Riemannian structures and positive densities. Example 6. Let A be a Lie subgroup of a Lie group G. Considering the action L of G on G / A (see Example 5 of 1.2), we can associate to any principal bundle G -+ E -+ B a certain bundle G / A -+ E -+ B with fibre G / A and structure group G. It turns out (see Husemoller 1966) that there exists a map

0: r(E)

-+

H1(B,FA)

such that the sequence

r(E) A

(j

-+

1

H (B,FA)

i* --+

1

H (B,Fa),

where i : A -+ G is an inclusion, is exact in the following sense: 1m 0 is the set of all fibre bundles with structure group A which can be obtained from E by reduction of structure group. Example 7. Let M be an n-dimensional manifold, A a Lie subgroup of GLn(lR). Let us denote by ~A the fibre bundle with base M and fibre GLn(lR)/A associated with the frame bundle R(M). The bundle ~A is known as the bundle of A-structures and its sections A-structures on M. As we saw in Example 6, to every A-structure there corresponds a reduction of the structure group of the bundle T(M) to the subgroup A. Many classical structures on manifolds are A-structures. In particular, for A = On A-structure is a Riemannian structure on M, for A = {g E GLn(lR) 1 det 9 = ±1}

a positive density, for A

= GL~(lR) = {g

E

GLn(lR) 1 det 9 > O}

an orientation on M, for A

= {g E GLn(lR) IgJ = Jg}

where J is some complex structure in lRn and n is even - an almost complex structure on M, and for A = {e} an absolute parallelization. 3.3. G-bundles. Let G be a Lie group and B be a differentiable G-space. By a G-bundle with base B we mean a differentiable bundle p : E -+ B together with a given G-action on E such that p is a morphism of actions. If E is a G bundle, then for any point x the stabilizer G x sends the fibre Ex to itself so that Ex is a Gx-space. Further, on the set of sections r(E) we have the following action of G:

(gs)(x) = g(S(g-lX)

(g

E G,

s

E

r(E), x

E

B).

(16)

If E is a vector bundle, then we shall assume that the mappings of fibres Ex -+ Egx(x E B) defined by the elements 9 E G are linear.

II. Lie Transformation Groups

119

Example 8. Let M be a differentiable G-space. Then the tangent bundle T(M) has a natural structure of a G-vector bundle: gv

= (dxTg)v (g

E G,v E Tx(M)).

The corresponding linear representation of G on the space tl(M) = r(T(M)) have the form g f--T (Tg ). (cf. (10)). Analogous G-structures are defined on the frame bundle, on the tensor bundles TP,q(M), on the bundles of A-structures for any Lie subgroup A C GLn(IR), n = dimM. A generalization of the situation described in Example 8 gives the following Lemma 3.1. Let P --t M be a principal bundle with structure group K, possessing the structure of a G-bundle such that the actions of G and K on P commute, i.e. g(xk) = (gx)k for all g E G,k E K,x E P. Then every bundle associated with P has a natural G-structure. Proof. Let E = P X K F be a bundle associated with P, the fibre of which is a differentiable K -space F. Then the action of G on E considered in Lemma 3.1, is given by the formula gK(x, y) = K(gx, y)

(g E G, x E P, y E F).

o

Example 9. If G is a Lie group, Band F differentiable G-spaces, then by the trivial G-bundle with base B and fibre F we mean the trivial fibre bundle E = B x F, with the following G-action: g(x,y) = (gx,gy)

(g E G,x E B,y E F).

Invariant sections of this G-bundle can be identified with morphisms of Gspaces B --t F. If F = lR. and G acts trivially on F, then the space of sections r(E) can be identified with the space of functions F(B) on the base B, and the action (16) with the representation PT considered in 1.5. An important problem in the theory of G-bundles is to describe the set r(E)G of invariant sections of a given bundle E. Examples of such sections are provided by G-invariant tensor fields, Riemannian structures and invariant densities on differentiable G-spaces (see Example 8). This problem will be considered in §3 of Chap. 2 in the case of a homogeneous space B and in Chap. 3 in the case of compact groups G. Another interesting problem is the question of the existence of a G-bundle structure on a given fibre bundle E --t B, whose base B is acted upon by a group G, i.e. the problem of lifting the action of G from B to E. For this problem see 1.6 of Chap. 3 (where the group G is assumed to be compact). 3.4. Induced Bundles and the Classification Theorem. In the theory of fibre bundles an important role is played by the following construction. Suppose we are given a differentiable fibre bundle p : E --t B with fibre F and structure group A. Then to any differentiable mapping f : M --t B there corresponds a fibre bundle E = f* E with base M, fibre F and structure group A, which is known as the bundle induced by f. It can be constructed by transferring

120

V. V. Gorbatsevich, A. L. Onishchik

the covering and the co cycle defining E to the manifold M. The total space E can be defined by the formula

E = {(x, y) EM x E I f(x) = p(y)}, projection p : E - t M by the formula p(x, y)

and the = p(y). One can view the induced bundle as a fibre bundle over M whose fibre over a point x E M is the fibre Ef(x) of the bundle E. Note that the mapping 7 : E - t E, given by the formula 7(x,y) = y takes Ex = {x} x Ef(x) to Ef(x). If G is some Lie group, E a G-bundle and f a morphism of G-spaces, then the action g(x,y) = (gx,gy) ((x,y) E E) turns E into a G-space, and 7 into a morphism of G-spaces. It turns out that any fibre bundle with structure group A and a given fibre F can be obtained by inducing it from a certain standard (so called universal) bundle, which depends only on A, F and the dimension of the base. Below we shall state the relevant theorem for the case of vector bundles. Example 10. Let V be a finite-dimensional vector space over one of the fields k = JR., C or 1Hl. Let Gm(V) denote the Grassmann manifold of all possible m-dimensional subspaces of V (1 :S m < dim V). The manifold E = {(1', x) E G m(V) x V Ix E 1'} is the total space of a differentiable vector bundle over k with' base Gm(V), m-dimensional fibre and projection (,)"x) f--+ 1'. This bundle is sometimes called tautological, because its fibre over point l' E Gm(V) is l' viewed as a vector space. The total space of the associated principal bundle is the manifold of all m-frames in V. We shall write G~(kn) = G~,m' The tautological vector bundle over G~,m can be viewed as a bundle with structure group On (k = JR.), Un (k = q or SPn (k = 1Hl). In this case the total space of the principal bundle is the Stiefel manifold St~ m of all orthonormal m-frames in kn. Note also that the tautological bundle has a natural structure of a GL (V)-bundle, with a transitive action of GL (V) on its base Gm(V). The following theorem shows that tautological bundles are universal for all vector bundles. Theorem 3.1 (See Husemoller 1966, Steenrod 1951). Any differentiable vector bundle over k with an m-dimensional fibre and an n-dimensional base B is induced from the canonical bundle by means of some differentiable map B - t G~,m' N 2: n + m + 1, ~ + m, n4"2 + m for k = JR., C, IHl respectively. Under these conditions two differentiable mappings B - t G~ m induce isomorphic vector bundles over B if and only if they are homotopi~.

In 1.5 of Chap.3 we shall consider a generalization of this theorem to G-bundles in the case when G is a compact Lie group.

II. Lie Transformation Groups

121

Chapter 2 Transitive Actions §l. Group Models 1.1. Definitions and Examples. Recall that any transitive action of a group G on a set M is isomorphic to an action of this group by left translations on the set of all cosets G I H where H = G;]) is the stabilizer of any point x EM. In the differentiable case (see 1.4 of Chap. 1) G I H is an analytic homogeneous space of the Lie group G and the isomorphism is a diffeomorphism. The Gspace G I H is called a group model (or a Klein model) of the homogeneous space M. A group model depends on the choice of the point x so that the subgroup H is defined up to conjugation in G. By means of a group model any property of the homogeneous space M can be expressed in terms of the group G and a subgroup H. We shall now illustrate this method on the simplest examples. Since all stabilizers of a transitive group G are conjugate, the kernel N of an action of G on M (which coincides with the intersection of all the stabilizers) is the largest normal subgroup of the group G contained in H. In particular, an action is effective if and only if H does not contain nontrivial normal subgroups of G and is locally effective if and only if H does not contain non-trivial connected normal subgroups of G. If we transfer from the given action of G to the effective action of GIN (respectively locally effective action of GINO), then a group model of the new homogeneous space will have the form (GIN)/(HIN) (respectively (GINO)/(HINO)). Next we give a description of morphisms of homogeneous spaces. Theorem 1.1. Let f : M --t N be a morphism of G-spaces, where M is homogeneous. Then f is surjective if and only if N is homogeneous. Group models of the homogeneous spaces M and N can be expressed in the form G I Hand G I K respectively, where K :J H, and then the morphism f takes the form gH f--? gK. Corollary. Homogeneous G-spaces G I HI and G I H2 are isomorphic if and only if the subgroups HI and H2 are conjugate in G.

Conversely, if G :J K :J H are inclusions of subgroups of G, then the natural mapping f : gH f--? gK of the homogeneous space G I H into G I K is a morphism. In the case when G is a Lie group and K and H are Lie subgroups, f is a projection of an analytic G bundle with fibre K I H and structure group K, associated with the principal bundle G --t GIK. Lemma 1.1. Let G be a Lie group and N :J H two Lie subgroups, where H is normal in N. Then the natural mapping f : G I H --t GIN is the projection of an analytic principal bundle, with structure group NIH, whose right action

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on G / H is given by the formula (gH) (nH) = gnH

(1)

(g E G, n EN).

Example 1. Let G be one of the classical compact Lie groups On (n ~ 2), Un (n ~ 1) or SPn (n ~ 1), acting linearly on the spaces JRn, Cn and lHIn respectively. Then G acts effectively and transitively on the unit sphere sn-l, s2n-l and s4n-l of the corresponding vector space. Taking as x the vector el of the standard basis we obtain the following group models G / H:

sn-l

= On/On-I,

s2n-l

= Un/Un-I,

s4n-l

= Sp/SPn_l.

In each of these cases the normalizer Nc(H) is the subgroup of all transformations in G preserving the line (el), so that G /N c (H) is a group model of the projective space over the corresponding field: IRP n - l = On/Ol x On-I, cpn-l = Un/U l x Un-I, lHIP n- l = SPn/SPl X SPn-l. The corresponding principal bundles sn-l --) IRPn-l, s2n-l --) cpn-l, s4n-l --) lHIP n- l with structure groups 0 1 ~ ~, U l , SPI respectively are known as Hopf bundles.

Example 2. Generalizing Example 1, consider the natural transitive action of the group G = On, Un, or SPn on the Stiefel manifolds St~ m (k = JR, C, 1HI respectively). Taking as x the frame {el, ... ,em} we obtai~ the following group models G / H j

St~,m = On/On-m,

St~,m = Un/Un- m, St~,m = Spn/Spn - m·

The subgroup Nc(H) consists of all transformations preserving the subspace (el, ... , em). Hence G/Nc(H) is a group model of the Grassmann manifold G~,m: G~,m = On/Om X On-m, G~,m = Un/Um X Un- m

G~,m = Spn/SPm

X

SPn-m.

The principal bundles St~ m --) G~ m have structure groups Nc(H)/ H ~ Om, Um, SPm respectively and are ass~ciated with the tautological bundles over G~ m (see Example 9 of 3.4 of Chap. 1). The following criterion for similarity of two transitive actions is easy to prove. Lemma 1.2. Let Ml = Gd HI and let M2 = G 2/ H2 be two homogeneous spaces. The effective actions of G l and G 2 on Ml and M2 are similar if and only if the pairs (GI, HI) and (G 2, H 2) are isomorphic, i.e. there exists an isomorphism ¢ : G l --) G 2 such that ¢(Hl ) = H 2 . 1.2. Basic Problems. We shall consider here some basic problems of the theory of homogeneous spaces and interpret them in the language of group models.

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123

Problem 1. Classify (up to isomorphism or similitude) all transitive actions of a given Lie group G. By Corollary 1 of Theorem 1.2 of Chap. 1, Corollary 1 of Theorem 1.1 and Lemma 1.2 Problem I is equivalent to the problem of classification of all Lie subgroups of G up to conjugation or up to automorphism of G. Many results have been obtained in the direction of such a classification, in particular in the case of connected subgroups of a connected compact or semi-simple Lie group (see volume "Lie Groups 3"). A differentiable manifold M is called homogeneous if M possesses a transitive action of some Lie group. Thus the notion of a homogeneous manifold differs from the notion of a homogeneous space in that in the first case there is no fixed Lie group action. The same homogeneous manifold may possess effective transitive actions of various Lie groups. For example on the sphere s2n-l (n ~ 2) there are transitive actions of the groups 02n, S02n, Un, and SUn, and on the sphere s4n-l (n ~ 1) of the groups 04n, S04n, U 2n , SU 2n and SPn (see Example 1). Problem II. Let M be a homogeneous manifold. Classify (up to isomorphism or similitude) effective actions on M of those Lie groups for which such actions exist. The solution to Problem II is known for a number of series of homogeneous spaces, for example for spheres (see Chap. 5 and 6). It reduces to the following question. Let M = G I H, where G is a Lie group, H a Lie subgroup, and let A be another Lie group. Find Lie subgroups B c A such that the manifolds M and AlB are diffeomorphic. If in Problem II we consider not only differentiable but also continuous actions, then there arises the problem: when is M homeomorphic to AlB? During investigation of these problems by means of homotopy invariants there arise naturally also the problems of classifying homogeneous spaces up to homotopy equivalence or rational homotopy equivalence. We also note a special case of Problem II, which consists of describing reductions and enlargements of a given transitive action (for more details see §4).

1.3. The Group of Automorphisms. We shall describe the group Auta M of all automorphisms of a homogeneous G-space M. Let H = G x , where x E M. Then formula (1) (where N = Na(H)) defines a right action of the group Na(H)1 H on M which is free and, in particular, effective. Theorem 1.2. The image of the right action of the group Na(H)1 H on M = GIH given by formula (1), coincides with the group Auta M. Thus AutaM S:' Na(H)IH. The group AutaM acts on M freely, and any of its orbits (AutaM)( x) coincides with the set Mare of those points y E M for which G y = G x .

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A transitive action of a group G on M (or the homogeneous G-space

M) is called asystatic if AutcM = {e} and systatic otherwise. It follows from Theorem 1.2 that an action is asystatic if and only if Nc(H) = H or

MH = {x}. Suppose now that M is a smooth homogeneous space of a Lie group G. Then N c (H) is a Lie subgroup of G and the group N c (H) / H is naturally a Lie group. We give AutcM the structure of a Lie group via the isomorphism of Theorem 1.2. By Lemma 1.1 the natural mapping M = G/H ---+ G/Nc(H) is the projection of a principal bundle with structure group AutcM. In the differentiable case it is convenient to introduce the notion of an asystatic action in a somewhat different way than for abstract groups. Namely, we shall say that a transitive differentiable action of a Lie group G on M is called asystatic if the group Autc M is discrete, or equivalently, if any point x E M possesses a neighbourhood U in M such that MC~ n U = {x}. For example, the standard action of the group SOn on sn-l (n 2': 2) is asystatic in the differentiable sense and systatic in the abstract sense (see Example 1). The corresponding actions of the groups Un and SPn on s2n-l and s4n-l are systatic even in the differentiable sense. The actions of these groups on projective spaces and Grassmann manifolds (Example 2) are asystatic in the abstract sense.

1.4. Primitive Actions. A transitive action of an abstract group G on a set M is called primitive if the only G-invariant equivalence relations ReM x M are R = {(x, x) I x E M} and R = M x M. In the language of a group model G / H, primitiveness is equivalent to H being a maximal proper subgroup. From Theorem 1.1 it follows that a transitive action is primitive if and only if any of its morphisms into a non-trivial transitive action is an isomorphism. For Lie groups primitiveness of an action is defined somewhat differently. A transitive action of a Lie group G on a manifold M is called primitive, if on M there is no G-invariant foliation with connected fibres of positive dimension smaller than dim M. This is equivalent to the following condition on the stabilizer H: for any virtual (see Part I) Lie subgroup fI such that H c fI c G we have fIo = HO or fIo = GO. Clearly every action of a Lie group which is primitive as action of an abstract group is also primitive in this sense. In what follows we shall consider the case when G is a connected Lie group and the action is locally effective. A Lie subgroup H eGis said to be primitive, if G acts primitively on M = G / H. A sub algebra of the tangent algebra 9 of G is called primitive, if it is the tangent algebra of some primitive Lie subgroup of G. Primitive subalgebras can be characterised as follows.

Theorem 1.3 (Golubitsky 1972). Let IJ be a subalgebra of a Lie algebra 9 not containing its ideals of positive dimension, and let H be the corresponding connected virtual Lie subgroup of G. The following conditions are equivalent: a) the subalgebra IJ is primitive;

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125

b) NG(H) is a maximal Lie subgroup ofG, with NG(H)O = H; c) I) is maximal among subalgebras not equal to 9 subalgebras of 9 which are invariant under AdNG(H). If the subalgebra I) is primitive then the group G acts primitively on the homogeneous space GIN G (H).

Clearly every maximal sub algebra of an algebra 9 which does not contain its non-zero ideals is primitive. Under certain conditions the converse statement also holds. Namely, let I) be a primitive subalgebra of a complex algebra g. If 9 is not simple or if 9 is simple and I) is non-reductive, then I) is maximal (Golubitsky 1972). At the same time, any simple complex Lie algebra contains a reductive primitive subalgebra which is not maximal (see Golubitsky 1972, Golubitsky and Rotschild 1971, where all such subalgebras of maximal rank are computed.) The simplest example is as follows. Example 3. Let 9 = s[n(C), n ~ 2, I) the subalgebra of all diagonal matrices with trace O. Then I) is primitive in g. To the pair (g, I)) there corresponds a primitive action of the group G = SLn(C) on the homogeneous space G ING(H), where H is the subgroup of all diagonal matrices in SLn(C). Clearly the sub algebra I) is not maximal. The classification of all reductive primitive subalgebras of complex or real simple Lie algebras is given in Komrakov 1991.

§2. Some Facts Concerning Topology of Homogeneous Spaces In this section we shall give a short survey of facts concerning coverings and real homotopy invariants of Lie groups and homogeneous spaces, which we shall use later. 2.1. Covering Spaces. Below we give a description of all coverings of a given connected homogeneous space in terms of group models. Let M be a connected homogeneous space of a Lie group G, According to Theorem 4.3 of Chap. 1 of Part I, the connected component of the identity GO acts transitively on M. We shall, therefore, suppose that G is connected and set H = G OJo, where Xo EM. Let 7r : G --> G be the universal covering of the group G. We have a transitive action (g,x) ~ 7r(g)x of the group G on M, with GOJo = H = 7r- I (H). By Theorem 4.8 of Chap. 1 of Part I, the manifold M = if I iIo = M is simply connected. By Lemma 1.1 the natural mapping p : if --> GIiI = M is the projection of a principal bundle with discrete structure group iII iIo, i.e. a covering. Thus p is the universal covering of the manifold M and 7r1 (M) ~ iII iIo. Theorem 2.1. Let M = G I H = GIiI, where G is the universal covering of a connected Lie group G. Then any covering manifold MI of M is a homogeneous space of the group G with group model GI HI, where iIo C HI C iI

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V. V. Gorbatsevich, A. L. Onishchik

and the covering Ml gH1 f--+ gil.

=

=

G / HI .....-) G / il

M coincides with the mapping

Proof. The proof follows from the fact that any covering Ml .....-) M is the projection of a bundle associated with the universal covering p : if . . .-) M, the fibre of which is a homogeneous space of the group 11"1 (M) ~ il / ilo. The action of G on Ml exists in view of Lemma 3.1 of Chap. 1. 0 2.2. Real Cohomology of Lie Groups. Let M be a manifold and A an associative commutative ring with unit. By H*(M; A) = ffip::::o HP (M, A) we denote the graded cohomology algebra of of the manifold M with values in A. Every continuous mapping f : M .....-) N generates a homomorphism of graded algebras f* : H* (N, A) .....-) H* (M, A). Let E = ffip::::oEp be a graded vector space over some field, where the dimensions bp = dim Ep are finite. We define the Poincare series of E by the formula: P(E, t) = bptp.

L

P::::O

In particular we have the Poincare series ofthe space H*(M,JR), where M is a manifold with finite Betti numbers bp(M) = dimHP(M,JR); we shall briefly denote it by P(M, t) (it is called the Poincare polynomial of the manifold

M).

Let G be a connected Lie group. Then (see Borel 1953) the real cohomology algebra H*(G;JR) of the manifold G is the exterior algebra A(6,.·· '~r)' where ~i are elements of odd degree 2mi + 1. The numbers mi (i = 1, ... ,r) are known as the exponents of G; they completely determine the algebra H*(G; JR). The largest exponent is denoted by m(G). Ifr = 0, i.e. H*(G, JR) ~ JR, we set m( G) = -1. The Poincare polynomial of G has the form

II(1 + em; +1 ). r

P( G, t) =

i=1

We set H+(G,JR) = ffip>oHP(G,JR) and denote by (H+(G,JR))2 the set of elements of the form ~:=1 UiVi, where Ui, Vi are elements of positive degrees. Then E(G) = H+(G, JR)/(H+(G; JR))2 is a graded vector space with a homogeneous basis ~i = ~i + (H+(G, JR))2 (i = 1, ... , r). We have

P(E( G), t) =

e

m1 +1

+ ... + t 2mr +1.

If G = G1 X G 2 is a direct product of two Lie groups, then H*(G,JR) H*(Gl, JR) ® H*(G 2, JR), whence E(G) ~ E(G 1 ) ffi E(G 2). Note that a finite covering G .....-) G generates an isomorphism H*(G,JR) ~ H*(G,JR) and that the cohomology of a connected Lie group G coincides with the cohomology of a maximal compact subgroup. From the above it follows that it suffices to know the exponents of simple compact connected Lie groups, which need only be considered up to local isomorphism. These exponents are given in Table 1 (see Bourbaki 1968).

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II. Lie Transformation Groups Table 1

G

r

ml···,mr

m(G)

SUn (n ~ 2) S02n+1 (n ~ 1) S02n (n ~ 3) SPn (n ~ 1)

n-1 n n n 6

1,2, ... , n - 1 1,3, ... ,2n - 1 1,3, ... ,2n - 3 1,3, ... ,2n - 1

n-1 2n-1 2n-3 2n-1

1,4,5,7,8,11 1,5,7,9,11,13,17 1,7,11,13,17,19,23,29 1,5,7,11 1,5

17 29 11 5

0

0

E6 E7 Es

F4 G2 S02

7 8 4 2 1

11

If G is a compact Lie group or a reductive algebraic group, then the exponents ml, ... ,mr of G possesses the following properties: 1. The number of exponents r = dim E{ G) coincides with the rank rk G of the group G i.e. with the dimension of a maximal torus T. 2. The numbers mi + 1 are the degrees of free generators of the polynomial algebra on the tangent space of the torus T, which are invariant under the Weyl group G. 3. If G is simple and non-commutative, then the number m{ G)+ 1 coincides with the Coxeter number of G, i.e. the order of a product of reflections in simple roots; moreover m{ G) = d:::'f - 2. 4. The number dimE{Gh (i.e. the number of exponents equal to 0) is equal to the dimension dim Z (G) of the centre of the group G. 5. The number dimE{Gh (i.e. the number of exponents equal to 1) is equal to the number of distinct non-commutative simple summands in the factorization of G into simple components .. Note also the following monotoneity property of the number m{G) (see Onishchik 1962, Onishchik 1979): Lemma 2.1. If H is a connected virtual Lie subgroup of a connected Lie group G, then m{H) :s; m{G). 2.3. Subgroups with Maximal Exponent in Simple Lie Groups. We shall call a connected virtual Lie subgroup H of a connected Lie group G a subgroup of maximal exponent if m{H) = m{G). Below we shall enumerate all such subgroups of simple non-commutative complex or compact Lie groups G. Theorem 2.2 (Onishchik 1962). Let G be a connected simple non-commutative complex or compact Lie group, H a proper connected virtual Lie subgroup (respectively real or complex) of maximum exponent. In the compact case the pairs H c G are exhausted by the following list, which contains one representative of a group G with a given simple tangent algebra, and H is

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V. V. Gorbatsevich, A. L. Onishchik

given up to conjugation in G:

SPn C SU2n

G2 c S07,

(n> 1),

S02n-l C S02n

(n> 3),

In the complex case the pairs H C G are obtained by complexification of the compact pairs listed above, i.e. are exhausted by the following list (under the same assumptions on G and H):

SP2n(C) C SL2n (C) (n> 1), G2(C) c S07(C), S02n-l(C) C S02n(C) (n > 3), Spin7(C) C SOs(C), G 2(C) c SOs(C), F 4 (C)

c

E6(C).

Corollary. Every connected subgroup H of maximal exponent in a connected simple complex or compact non-commutative Lie group G is simple, closed in G and coincides with the connected component of its normalizer. We have rkH < rkG. If we exclude the case G 2(C) c SOs(C), then the tangent algebra of such a subgroup is maximal in g. 2.4. Some Homotopy Invariants of Homogeneous Spaces. We shall next consider some invariants of the real homotopy type of a homogeneous space, which can be easily expressed in the language of group models. Invariants of this kind are particularly useful in solving problems of classification of homogeneous spaces (see 1. 2). We shall denote by r s (M) the rank r k 11"8 ( M) ofthe s-th homotopy group of a manifold M (for s = 1 this number is defined only when 1I"1(M) is abelian). Let h : G 1 ---+ G 2 be a homomorphism of Lie groups. Then the corresponding homomorphism of cohomology algebras h* : H*(G 2,1R) ---+ H*(Gl,IR) induces a homomorphism of graded vector spaces E(G 2) ---+ E(G 1 ) which we shall also denote by h *. Theorem 2.3 (Onishchik 1963). Let H be a connected Lie subgroup of a connected Lie group G, M = G / H and let i : H ---+ G be the inclusion map. Set Eo(G) = Keri*, EO(H) = Cokeri*, where i* : E(G) ---+ E(H). Then dimEo(Ghk+1

= r2k+1(M) (k

dimEo(Hhk+l = r2k+2(M)

~

0),

(k ~ 0).

Corollary 1. The polynomial 00

P(E(G), t) - P(E(H), t) = ~)r2k+1 - r2k+2)t 2k +1 k=O

is a homotopy invariant of the manifold M

= G / H.

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II. Lie Transformation Groups

We define the homotopy characteristic of a manifold M by 00

h(M)

= - ~) -1)8 r8 (M). 8=1

By the rank and co rank of a manifold M we mean the numbers defined respectively by the formulas 00

rkM

=

L r2k+l,

00

corM = Lr2k. k=1

k=o

Then h(M) = rkM - cor M. Corollary 2. We have

rkM = dimEo(G), cor M = dimEoOH), h(M)

In particular,

= dimE(G)

o :::; cor M

- dimE(H) :::::

o.

:::; rk M,

0:::; h(M) :::; rkM. From Corollary 2 it follows that if G and H are compact, then rkG - rkH = h(M) is a homotopy invariant of the manifold M

= G / H.

Lemma 2.2. Let G be a connected compact Lie group and H a Lie subgroup, then rk (G / H) = 0 if and only if the manifold G / H is diffeomorphic to ]R.n.

Let X(G/H) = P(M, -1) be the Euler characteristic of the manifold M. We have the following assertion (see Samelson 1958): Lemma 2.3. Let G be a connected compact Lie group and H a Lie subgroup ofG. Then X(G/H)::::: 0 with X(G/H) > 0 if and only ifrkG = rkH.

Note that X(M) ::::: 0 for any compact homogeneous manifold M (see Corollary 1 of Theorem 1.2 of Chap. 5). It is known that X(M) = 0 for a compact differentiable manifold M if and only if on M there is a non-vanishing COO-vector field. Hence we have the following Lemma 2.4 (Hermann 1965). If M is a compact differentiable manifold and X(M) #- 0, then any transitive action of a Lie group on M is asystatic.

§3. Homogeneous Bundles Let G be a Lie group. By a homogeneous bundle with respect to G (or a homogeneous G-bundle) we mean a G-bundle the base of which is a homogeneous space of G. Many important geometric structures on homogeneous

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V. V. Gorbatsevich, A. L. Onishchik

spaces are sections of homogeneous bundles. In particular we note that homogeneous vector bundles play an important role in the theory of representations of Lie groups. 3.1. Invariant Sections and Classification of Homogeneous Bundles. Let M be a homogeneous space of a Lie group G, x E M and H = G.,. Let P : E --+ M be a G-bundle with base M and let F = E.,. Then F is a differentiable H-space. Consider the set r(E) of all differentiable sections of a bundle E and its subset r(E)G of sections invariant under the action given by formula (16) of Chap. 1, i.e. morphisms of G-spaces s : M --+ E. Let r., : r(E) --+ F be the mapping given by the formula rz(s) = s(x). Clearly, r z is H-equivariant, whence it follows that r z takes r(E)G to FH. Since the action of G on M is transitive it follows that the mapping is invertible. The inverse mapping sends an element U E FH into a section s E r(E)G given by the formula s(gx)

= gu (g

E G).

Thus we have Theorem 3.1. For any homogeneous bundle F r z : r(E)G --+ FH is bijective.

--+

E

--+

M the mapping

Analogously one describes morphisms of a homogeneous bundle E into p

another homogeneous bundle F' ~ E' ~ M', i.e. morphisms of G-spaces h : E --+ E' such that p' 0 h = p. The morphism h is called an isomorphism if it has an inverse morphism. Any morphism h : E --+ E' defines a morphism of H -spaces hz : F --+ F' = E~. Theorem 3.2. The correspondence h f--+ hz is a bijection between the set of morphisms of homogeneous bundles E --+ E' and and the set of morphisms of H -spaces F --+ F'. Moreover, h is an isomorphism if and only if hz is an isomorphism for each x.

From this theorem we can deduce the following theorem on classification of homogeneous bundles with base M = G / H. Theorem 3.3. By assigning to each G-bundle E --+ M the differentiable H -space F = Ez we obtain a one to one correspondence between isomorphism classes of homogeneous bundles with base M and isomorphism classes of differentiable H -spaces. Proof. The existence of a homogeneous bundle E to which a given Hspace F corresponds can be proven as follows. The group G can be viewed as a principal bundle with base M and structure group H (see Example 3 of 3.2 of Chap. 1), which is homogeneous with respect to the action by left translations. By Lemma 3.1 of Chap. 1 the associated bundle E = G XH F is also homogeneous. 0

II. Lie Transformation Groups

131

From Theorem 3.3 it follows that every homogeneous bundle over G / H admits H as a structure group. Note that the classification Theorem 3.3 is finer than the usual classification of bundles in the sense of differential topology, since it takes into account the action of G. In particular, there exist trivial bundles which are not isomorphic as G-bundles. According to Theorem 3.3, two trivial homogeneous spaces (Le. trivial G-bundles over M, see Example 8 of 3.3 of Chap. 1) are isomorphic if and only if the reductions to H of the actions of G on F which define them, are isomorphic. Theorem 3.4. A homogeneous G-bundle F -+ E -+ M is isomorphic to a trivial bundle if and only if the action of the group of H on F = Ex admits an enlargement to an action of G. 3.2. Homogeneous Vector Bundles. The Frobenius Duality. A homogeneous vector bundle is a G-vector bundle over a homogeneous space G / H. Recall, (see 3.3 of Chap. 1), that the action G is linear on fibres. In particular, the action of the group H on the space F = Ex is linear. We shall also suppose that morphisms of homogeneous vector bundles are linear on fibres. Morphisms of homogeneous vector bundles E -+ E' form a vector space HomG(E, E'). Let us formulate analogues of Theorem 3.1-3.3 for homogeneous vector spaces (the analogue of Theorem 3.4 is, of course, also valid). Theorem 3.5. For any homogeneous vector bundle F -+ E -+ M the map rx : r(E)G -+ FH is an isomorphism of vector spaces. If F' -+ E' -+ M is another homogeneous vector bundle, then HomG(E', E) ~ HomH(F', F). Theorem 3.6. Assigning to each homogeneous vector bundle E -+ M the linear H -space F = Ex we obtain a one to one correspondence between isomorphism classes of homogeneous vector bundle with base M and isomorphism classes of finite-dimensional linear representations of the Lie group H.

Thus to each linear representation p : H -+ GL (F) on a finite-dimensional vector space F there corresponds a homogeneous vector bundle Ep with base M = G/H. We denote by R: G -+ GL(r(Ep)) the linear representation of the group G on the space of differentiable sections of the bundle E given by formula (16) of Chap. 1 (and, generally speaking, infinite-dimensional). The representation R is called the representation of G induced by the representation p of the subgroup H. Study of induced representations is based on the following classical result. Theorem 3.7. For any linear G-space V and any linear representation p: H -+ GL(V) we have an isomorphism of vector spaces

Proof. The statement follows from the Theorem 3.5 if we take as E' the trivial bundle M xV. 0

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V. V. Gorbatsevich, A. L. Onishchik

Note that, if we identify Ep with G XH F, then r(Ep) can be identified with the space of differentiable vector-functions ¢ : G -+ F such that ¢(gh) = p(h)-l¢(g). The induced representation R acts on this space by the formula (R(g)¢)(x)

= ¢(g-lx)

(g, x E G).

3.3. The Linear Isotropy Representation and Invariant Vector Fields. Let M = G / H be a homogeneous space. As we saw in Example 7 of 3.2 of Chap. 1, the tangent bundle T(M) admits the structure of a G-vector bundle, i.e. homogeneous vector bundle with base M. The corresponding representation of the subgroup H is the linear isotropy representation £ : H -+ GL(T",(M)) (see 1.3 of Chap. 1). We shall describe the representation £ in terms of the group model. Since the map dT: : 9 -+ T",(M) is surjective and its kernel coincides with ~ (see 1.3 of Chap. 1), the space T",(M) can be identified with g/~. Lemma 3.1. The isotropy representation £ can be identified with a quotient representation of the adjoint representation of the group H on the space g, i. e. £(h)(x

+ ~) =

Further (d£)(y)(x +~)

(Ad h)x + ~

=

[y, x]

+~

(h E H, x E g).

(3)

(y E ~,x E g).

(4)

A homogeneous space M = G / H is called reductive if there exists a subspace meg, such that 9 = m E9 ~ and (Ad H)m = m. For reductivity it is sufficient, for example, that H be a compact or semi-simple group with a finite number of connected components. The tangent space T",(M) to a reductive homogeneous space can be identified with m and the isotropy representation with the corresponding subrepresentation of the representation Ad of the subgroup H (see Helgason 1962). Let tl(M) be the space of all differentiable vector fields on a manifold M. From Theorem 3.5 and Lemma 3.1 follows Theorem 3.8. The subspace tl(M)G of all invariant vector fields on M is isomorphic to the space (g/~)H of invariants of the representation £ given by formula (3).

As is shown by the Corollary to Theorem 2.7 of Chap. 1, tl(M)G is a subalgebra of the Lie algebra tl(M) tangent to the Lie group AutGM, which by Theorem 1.2 is isomorphic to NG(H)/ H. The isomorphism of Theorem 3.8 sends the commutator in tl(M)G to the operation (x +~, y +~) I-t [x, y] + ~ (x + ~,y + ~ E (g/~)H). 3.4. Invariant A-structures. Below we retain the notation of 3.3. We shall find it convenient to identify the structure group of the frame bundle R( M) with the group GL(g/~). Let A be a Lie subgroup of GL(g/~). Then the bundle of A-structures EA -+ M is homogeneous (see Example 7 of 3.2 of Chap. 1 and Lemma 3.1). Our purpose is to describe the set of all G-invariant A-structures on M. Let Iso = £(H) c GL (g/~) be the linear isotropy group

133

II. Lie Transformation Groups

of the homogeneous space M. Let S = {g E GL(g/~) Ig- 1 (Iso)g c A}. Clearly the group A acts on S by right translations. From Theorem 3.1 we deduce Theorem 3.9. G-invariant A-structures on M = G / H are in one to one correspondence with elements of the set S / A. In particular such a structure exists if and only if S -I 0. Corollary 1. There exists on M = G / HaG-invariant Riemannian structure if and only if the closure of the linear isotropy group in GL(g/~) is compact. Such structures are in one to one correspondence with positive definite quadratic forms on the space g/~ which are invariant under L.

Proof. The proof makes use of the fact that for any action of a compact linear group there exists an inner product invariant under it (this easily follows from Lemma 3.2, see below). 0 Corollary 2. The homogeneous space M = G / H possesses a G-invariant orientation if and only if det L( h) > 0 for any h E H or, equivalently, if ~:!:!~~ > 0 for all h E H. If G is connected, then this condition is both necessary and sufficient for orientability of the manifold M. Example 1. Let M = G be a Lie group acting on itself by left or right translations. Then H = {e}. From Corollary 1 if follows that G has left invariant (i.e. invariant under all left translations) and also right invariant Riemannian structures; they are in a bijective correspondence with inner products in the tangent algebra g. Consider now the action of G x G on G by left and right translations. In this case L(g,g) = Adg (g E G), and the subgroup Iso coincides with the adjoint linear group Ad G. Corollary 1 shows that on any compact Lie group G there exists a bi-invariant (i.e. invariant under both left and right translations) Riemannian structure. Such a Riemannian structure is necessarily invariant also under the transformation 9 ~ g-l of G. If, in addition, G is simple, then the group Ad G is irreducible, whence it follows that all (Ad G)-invariant inner products in 9 are proportional. Thus, a bi-invariant Riemannian structure on a simple compact Lie group G is unique up to a positive scalar factor. Note that for a homogeneous space which admits an invariant Riemannian structure the linear group Iso does not necessarily have to be compact. This is demonstrated by Example 2. Consider the following action of the group G = SUn the complex Stiefel manifold M = St~,2 :

(g,s)(x,y) = (e27risg(x),e27ri8sg(y))

X

IR. on

(g E SUn,s E IR.,(x,y) E St~,2)'

where () is a fixed irrational number. One can easily verify that Iso is a virtual linear Lie group isomorphic to SUn - 2 X IR. and the subgroup Iso is compact.

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On the other hand, it is known that, if G is a Lie group of all automorphisms of a Riemannian structure on some manifold M, then the subgroup G;x (and therefore, also the linear isotropy group) is compact for any point x E M (see Ch. 3, 1.1, Example 3). Example 3. The Euclidean space En is a homogeneous space of its group of motions E(n), and moreover Iso = On. The latter is also true for the linear isotropy group of the sphere sn considered as a homogeneous space of the group On+1 (see Example 3 of 1.2 of Chap. 1). The same holds for the Lobachevski space An, which is defined as the connected component of the hyperboloid

SO,n

= {x

E IRn +11 X~

-

X~

-

••• -

x~+1

= I}.

given by the inequality Xl > 0 and is considered as a homogeneous space of the group G = Ol,n (see 4.2 of Chap. 1 of Part I). According to Corollary 1 each of these spaces has a unique (up to a positive scalar factor) G-invariant Riemannian structure. The spaces En (n 2: 1), sn (n 2: 2), An (n 2: 2) can be characterized as the simply-connected homogeneous spaces M of real Lie groups, satisfying one of the following conditions: there exists on M an invariant Riemannian structure a with constant sectional curvature; the linear isotropy group coincides with the full orthogonal group of the tangent space (with respect to some Euclidean metric). Moreover, under these conditions G = Aut (M, a) (see Wolf 1972). 3.5. Invariant Integration. Another consequence of Theorem 3.9 is the

following Theorem 3.10 There exists a G-invariant positive density on M = G / H if and only if det £(h) = ±1 for any h E H or, equivalently, if IdetgAd hi = Idet~Adhl for any h E H. A G-invariant density is unique up to a constant factor.

Example 4. On any Lie group G there exists a left-invariant and a rightinvariant positive density (each of them unique up to a positive constant factor). In order that a left-invariant density be right-invariant, or equivalently, in order that there exist a bi-invariant positive density on G it is necessary and sufficient that det Adg = ±1 for any 9 E G. Lie groups with these properties are called unimodular. Any of the following conditions is sufficient for unimodularity of a group G: the group Ad G is compact; the group G is semi-simple; the group G is connected and nilpotent. For any positive density on a manifold M there is an associated measure on M. The integral with respect to this measure of a compactly supported continuous function ¢ on M JM ¢( x) dx is defined as follows. Suppose that the support Supp ¢ is contained in a coordinate neighbourhood with local coordinates Xl,"" xn and let p be the coordinate of our density in this coordinate system. Then

135

II. Lie Transformation Groups

In general, let Supp ¢l c

s

U Ui , where Ui

are coordinate neighbourhoods and

i=l

let {ei Ii = 1, ... , s} be a partition of unity on

t 1M

covering (Ui ). Then

1M ¢lex) dx =

s

U Ui

corresponding to the

i=l

ei(x)¢l(x) dx.

If M is a smooth G-space and the density is G-invariant, then integration possesses the following property:

1M ¢l(gx) dx = 1M ¢lex) dx

(g E G).

(5)

Integration possessing property (5) can also be defined for functions ¢l with values in a Fnkhet space. Let G be a Lie group. A Frechet space S is called a topological G-module if there is given on S a structure of a topological linear G-space (or, equivalently, there is given a linear representation of G on S such that the mapping (g, s) I-t gs of the space G x S into S is continuous). The following lemma is frequently used in constructing invariants.

Lemma 3.2. Consider a left-invariant positive density on G. If S is a topological G-module and for some s E S the integral So = fG (gs) dg exists, then So E SG. In particular when G is compact and fG dg = 1, the operator I : s I-t So linearly and continuously transforms S onto SG, with I ISO = id. An example of a topological G-module is provided by the space r(E) of sections of a G-vector bundle with the COO-topology and the action of G given by formula (16) of Chap.l. A special case is the space F(M) of differentiable functions on a differentiable G-space with representation PT described in 1.4 of Chap. 1. In computing integrals on Lie groups and homogeneous spaces the following lemma is often useful

Lemma 3.3 (see Helgason 1962). Suppose that a homogeneous space G / H has a G-invariant positive density and let ¢l be a function on G which is integrable with respect to the left invariant positive density on G. Then after a suitable normalization of densities [ ¢leg) dg

lG

= [

where 1jJ(gH)

=

lG/H

1jJ(x) dx,

L

¢l(gh) dh

(this integral is taken with respect to the left-invariant positive density on H.)

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The statement of Lemma 3.3 can be reformulated as the following double integration formula:

fa ¢(g) dg = fa/H (L ¢(gh) dh) d(gH).

3.6. Karpelevich-Mostow Bundles. This subsection is devoted to a theorem which makes it possible to represent a homogeneous space G / H, where G and H have a finite number of connected components, as the total space of a homogeneous vector bundle, whose base is a homogeneous space of a maximal compact subgroup of the group G (the Karpelevich-Mostow bundle). This theorem generalizes the following property of Lie groups: any Lie group G with a finite number of connected components is diffeomorphic to the manifold K x ]Rn, where K is a maximal compact subgroup of the group G (see "Lie groups and Lie algebras - 3", Encycl. Math. Sc. 41, Ch. 4, 3.4). Let G be a Lie group, A and B two analytic submanifolds of G. We shall write G = A x B if the map (a, b) r-t ab of the manifold A x B into G is an analytic diffeomorphism. This relationship generalizes to any number of factors. The construction of the bundle sought is based on the following lemma (see Mostow 1955b). Lemma 3.4. Let G be a Lie group, Hand K a Lie subgroups, L = K n H, E and F analytic submanifolds in G, with G = K x F x E, H = L x E, e E F, and let gFg- 1 = F for all gEL. Then the mapping L(k,f) r-t kfH is a K -invariant analytic diffeomorphism of the manifold K XL F onto G/ H.

Let now G be a Lie group with a finite number of connected components, H a Lie subgroup with the same property, and K J L two maximal compact subgroups of these groups containing one another. It turns out (see Mostow 1955b, Mostow 1962b) that there exist submanifolds E,F C G satisfying the

conditions of Lemma 3.4 which are analytically diffeomorphic to Euclidean spaces. From this we obtain Theorem 3.11. Let H be a Lie subgroup of a Lie group G, where G and H have a finite number of connected components and K J L are maximal compact subgroups of Hand G respectively. Then G/ H as an analytic Kspace is isomorphic to some homogeneous vector bundle with base K / L.

In the case when G and H are connected and semi-simple, this theorem was proved independently by F.1. Karpelevich (1956) and Mostow (1955b), and in the general case by Mostow (1962b). In the semi-simple case the proof of the existence of submanifolds E, F follows from the existence of compatible Cartan decompositions for a semi-simple real Lie algebra and a semi-simple subalgebra (Karpelevich 1953, see Mostow 1955a). In order to grasp the scope of applicability of Theorem 3.11 we note that, if a Lie group G with a finite number of connected components acts transitively on a connected manifold M = G / H with a finite fundamental group, then H has a finite number of connected components. Another important case when the conditions of the

II. Lie Transformation Groups

137

theorem are satisfied is the case of an algebraic group G (over the field lR or C) and an algebraic subgroup H. Corollary 1. With the assumptions of Theorem 3.11 the manifold G I H has the homotopy type of the compact homogeneous manifold K I L. Corollary 2. If 7r8 (G I H) = 0 for all s 2: 0, then the manifold G I H is analytically diffeomorphic to lRn . Corollary 3 (Montgomery's Theorem). If under the assumptions of Theorem 11 the homogeneous space G I H is compact, then the maximal compact subgroup K ofG acts transitively on GIH, i.e. GIH = KIL. Going over to the universal covering group (see 2.1) we derive the following statement. Corollary 4. If a connected Lie group G acts transitively on a compact manifold M with a finite fundamental group 7r1(M), then the commutator of any maximal compact subgroup of G and a Levi subgroup (a maximal connected semi-simple subgroup) of G acts transitively on M. Corollary 5. If G and H are connected and dim G I H - dim K I L G I H is analytically diffeomorphic to K I L x lR.

= 1, then

Example 5 (Samelson, see Mostow 1955b). Consider the projective space ]Rpn with the standard projective action of the group GL n+1 (lR). Let Xo E ]Rpn and M = ]Rpn - {xo}. Then M is a homogeneous space of the group G = GLn+1 (lR)xo, with M = G I H, where H = GLn+l (lR)xo n GLn+1 (lR)Xll Xl E M. It is easy to see that K I L is naturally isomorphic to the hyperplane ]Rpn-l C ]Rpn and the projection of the Karpelevich-Mostow bundle M --; KI L sends each point x E M to the intersection point of the projective line passing through x and Xo with ]Rpn-l. The fibre of this bundle is onedimensional but it is non-trivial. Example 6. In (Mostow 1955b) there is also an example of a simplyconnected homogeneous space of a semi-simple Lie group for which the Karpelevich-Mostow bundle is non-trivial. In this example GIH = SL 3(C)/SL 3(lR), KI L = SU3/S03. Example 7. The conditions imposed on G and H in Theorem 3.11 are genuinely needed. Let, for example, G = SL 2(lR) and H be any uniform discrete subgroup of G. Then the action of the maximal compact subgroup K = S02 C G on G I H is non-transitive, so that Corollary 3 is in this case false.

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§4. Inclusions Among 'fransitive Actions In this section we consider the problem of describing all inclusions between transitive (effective) actions of connected Lie groups on a given homogeneous manifold. It can be viewed as a part of the problem of classification of all transitive actions (see 1.2). The problem of describing enlargements arises naturally, for example, in the following situation. Suppose that a Lie group G acts on a manifold M, on which there is given a certain G-invariant geometric structure a such that the group Aut (M, a) of all automorphisms of this structure is a Lie group (see 2.4 of Chap. 1). Then the natural action of the group Aut (M, a) is an enlargement of the given action of G. Hence knowledge of all enlargements of the group G helps to determine the group Aut (M, a). Studying inclusions between transitive actions is equivalent to studying factorizations of Lie groups as products of two Lie subgroups. We remark that of interest to physics is the problem of unification of two Lie groups, i.e. of constructing Lie groups which decompose as product of the given groups (see Sternheimer 1968, Barut and Rq,czka 1977). We shall return to the consideration of inclusions in Chapters 3 and 5. 4.1. Reductions of Transitive Actions and Factorization of Groups. It is clear that enlargements of a transitive action are always transitive. Next we shall explain when the same holds for a reduction of a transitive action of a group to a subgroup. Lemma 4.1. Let G be some group, A and H two subgroups. Then the following conditions are equivalent: a) the reduction of the natural action of G on G / H to A is transitive, b) G = AH; c) G = HA; d) the reduction of the natural action of G on G/ A to the subgroup H is transitive. Condition b) means that every element g E G can be represented in the form g = ah, where a E A, h E H. In this case we shall say that G decomposes as product of the subgroups A and H or that the triple (G, A, H) is a factorization of the group G. Corollary 1. A triple (G, A, H), where A and H are subgroups of G, is a factorization if and only if the triple (G, H, A) is one. Corollary 2. If (G, A, H) is a factorization, then· so is the triple (G, aAa- 1 , bHb- 1 ), where a, b are any elements of G. Corollary 3. The reduction of the natural action of G on G / H to a subgroup A eGis simply transitive if and only if G = AH (or G = H A) and An H =

{e}.

II. Lie Transformation Groups

139

Consider the action of the group G x G on G by two sided translations (see Example 4 of 1.2 of Chap. 1). The stabilizer of the point e EGis the diagonal G d C G x G, which is isomorphic to G. Applying Lemma 4.1 to the subgroup A x H c G x G we obtain Corollary 4. The following conditions are equivalent: a)G=AH; b) G x G = (A X H)Gd; c) G x G = Gd(A x H).

Suppose we have an inclusion between transitive actions T' :::; T (see 1.1 of Chap. 1) of groups G' and G on a set M. Clearly, Auta,M :J AutaM. Hence if the action T' is asystatic, then so is the action T. Analogously, an enlargement of a primitive action is primitive. 4.2. The Natural Enlargement of an Action. For any action T of a group G on M we denote by SimaM the group ofits autosimilitudes (Le. of similitudes of T onto itself). Clearly, SimaM contains AutaM as a subgroup. For any g E G the transformation Tg is an autosimilitude of T (the corresponding automorphism of G being the inner automorphism Ag), so that we have a homomorphism t : G -+ SimaM. Its image t(G) is a normal subgroup of SimaM commuting with AutaM elementwise. If T is effective we may identify G with t(G). We denote by N(T) the natural action of the group SimaM on M. If T is effective, we may consider N(T) as an enlargement of the action T; this enlargement will be called natural; it is proper if and only if AutaM i=- {id}. Next we describe the group SimaM in the case when the action T is transitive. Write H = Gzo ' where Xo E M. Let Aut (G, H) be the group of automorphisms of G mapping H onto itself. Any 0: E Aut (G, H) defines an autosimilitude Sa by

sa(gxO)

Then A = {sa

= o:(g)xo, g E G.

10: E Aut(G, H)} is a subgroup of SimaM fixing the point Xo.

Lemma 4.2. The subgroup A coincides with (SimaM)",o' The action T is transitive if and only if SimaM = t(G)A. Corollary. For the action L of a group G on M = G by left translations, the group SimaM coincides with HolG = l(G)(AutG) of the group G.

Using Lemma 4.2 and some properties of semi-simple Lie groups (see Lie groups and Lie algebras - 3, Encycl. Math. Sc. 41, Ch. 3, §3) we obtain Theorem 4.1. Let M = G / H, where G is a connected semi-simple Lie group acting effectively on M. Then SimaM has a finite number of components and (SimaM)O = G (AutaM)O (locally direct product). If G is compact, then SimaM is compact too, and the natural homomorphism Aut (G, H) -+ A is an isomorphism.

140

V. V. Gorbatsevich, A. L. Onishchik

Now we mention some applications of Lemma 4.2 and Theorem 4.2. Suppose that we have an enlargement T' of an action T of a group G on M, such that the corresponding group G' contains G as a normal subgroup. If T' is effective, then G' is identified with a subgroup of SimcM, so that T ::; T' ::; N(T). If, in addition, T is transitive, then we get a description of G' in terms of the group G and its automorphisms. More precisely, we have Corollary 1. Let G be a normal subgroup of G', T a transitive action of G on M, and T' an effective enlargement ofT to G'. Then G' can be identified with a subgroup ofSimcM of the form G' = GB, where B is a subgroup of A. If G is connected and semi-simple, then G'o = GC, where C c (AutcM)o. A transitive action of a group G on M is called irreducible if no proper normal subgroup of G acts on M transitively. Similarly, a transitive action of a connected Lie group G is called irreducible if no proper connected virtual Lie subgroup of G acts transitively. Corollary 1 permits us to describe transitive actions in terms of irreducible ones. Since the connected component of the identity of a Lie group G is normal in G, we get Corollary 2. Let T be a transitive effective action of a Lie group G on a connected manifold M. Identifying G with t(G) we have G C SimcoM and G = GO B, where B is a subgroup of A = (Simco M) Xo • 4.3. Some Inclusions Among Transitive Actions on Spheres. We now consider transitive actions of classical compact Lie groups on spheres, described in 1.1 in Example 1. Viewing the space en as lR.2n we obtain an inclusion Un C 02n, with both groups acting transitively on sn-I. Since sn-I (n 2 0) is connected, the subgroup SOn C On is transitive on sn-I. It is easy to show that the subgroup SUn C Un is transitive on s2n-l, so that we also have an enlargement of actions S02n =:J SUn on s2n-l. Further, considering the space IHIn as e 2n , we obtain an inclusion SPn C SU 2n , with both groups acting transitively on s4n-l. Since Auts Pn s4n-1 '::::: SPI' we also obtain a natural enlargement SPn C SPn X SPI' which in this case is locally effective. This enlargement is not a reduction of actions of groups U 2n and SU 2n but it turns out that there exists a homomorphism ¢ : SPn X SPI --+ SP4n' which defines an enlargement of actions on s4n-I. This homomorphism is given by the formula ¢(g,q)u

= guq-I

(g E SPn,q E SPI'U E W).

Thus we obtain the following scheme for enlargement of actions on sn-I, each of which can take place if the indices in the symbols denoting classical groups are integers: SPI! --+ SUI!2 --+ UI!2 4

1 SPI!4

1 X

SPI

--+

SOn

~

/

On

II. Lie Transformation Groups

141

We note the following factorizations of Lie groups, which follow, by Lemma 4.1, from the above enlargements:

S02n = SUn· S02n-l, S04n = SPn . S04n-b

SU 2n = SPn . SU2n - 1 , 4.4. Factorizations of Lie Groups and Lie Algebras. Lemma 4.1 shows that description of inclusions between transitive actions of Lie groups is equivalent to description offactorizations of the form (G, A, H), where G is a Lie group, A a virtual Lie subgroup and H a Lie subgroup. It is natural to consider the following, somewhat more general, notion. A factorization of a Lie group G is defined to be a triple (G, G', Gil), where G' and Gil are virtual Lie subgroups of G with G = G'G". We note some general properties of factorizations of Lie groups. First of all, by Corollary 1 of Lemma 4.1, the triple (G, G', Gil) is a factorization if and only the same holds for the triple (G,G",G'). From now on, two factorizations differing only in the order of summands will be considered identical. Further, for every triple (G, G', Gil), where G', Gil are virtual Lie subgroups of a Lie group G, there is a corresponding triple of Lie algebras (g, g', gil), where 9 ::J g', gil are the tangent algebras of the Lie groups G, G', Gil. Two triples of the form (G, G', Gil) are called locally isomorphic if the corresponding triples of tangent algebras are isomorphic.

Lemma 4.3 (Onishchik 1969). Let (G, G', Gil) and (G, G', Gil) be two locally isomorphic triples of Lie groups, with G and G' connected. Then G = G' Gil if and only if G = G' Gil . This lemma shows that in studying factorizations (G,G',G") of a connected Lie group G we can assume that G', Gil are connected, and also that we can replace G by any connected Lie group locally isomorphic to it. In this case the property of decomposability of a triple (G, G', Gil) depends only on the corresponding triple of tangent algebras. By a factorization of a Lie algebm 9 we mean a triple (g, g', gil) where g' and gil are subalgebras of 9 and 9 = g' + gil. Lemma 4.4. Let G be a Lie group, G', Gil a virtual Lie subgroups and

(g, g', gil) the corresponding triple of tangent algebms. Then the following conditions are equivalent: a) 9 = g' + gil ; b) 9 $ 9 = (g' $ gil) + gd, where gd is the diagonal in 9 $ g; c) the set G'G" is open in G. If Gil is closed in G then these conditions are equivalent to the condition d) the orbit G'(xo), where Xo = Gil E GIG", is open in GIG". In particular we see that a factorization of a Lie group determines a factorization of its tangent algebra. The converse statement is, in general, not true (see EXanlple 1 below).

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V. V. Gorbatsevich, A. L. Onishchik

We shall call a factorization (g, g', gil) global if G = G' Gil, where G is a connected Lie group with tangent algebra g. Note the following criterion of globality of factorizations. Lemma 4.5. A triple of algebras (g, g', gil) is a global factorization if and only if for any inner automorphism 0: of the algebra 9 we have 9 = o:(g') + gil .

We also note that a factorization of a Lie algebra is global if the following conditions are satisfied: G' is compact, Gil is closed in G. From this we deduce: Theorem 4.2 (Onishchik 1962). Any factorization of a compact Lie algebra 9 is global.

In (Malyshev 1975) it is shown that this property continues to hold for almost compact Lie algebras, i.e. semi-direct sums 9 e- a where 9 is compact and a is a real commutative Lie algebra, with a given orthogonal linear representation of the algebra g. For globality conditions for factorizations of reductive Lie algebras see 3.2 of Chap. 3. We also note the following result. Theorem 4.3 (Malyshev 1978). Any factorization of a nilpotent Lie algebra (over IR or i.C) is global. If every factorization of a complex Lie algebra 9 as a sum of complex subalgebras is global, then 9 is nilpotent.

The following simple lemma establishes a connection between factorizations of real and complex Lie algebras. Lemma 4.6. A triple (g, g', gil) of real Lie algebras is a factorization if and only if the same holds for the triple (g(C), g' (C), gil (C)).

For Lie groups only the "only if" part of the statement is valid. Lemma 4.7 (Onishchik 1969). Let (G, G' ,Gil) be a factorization, where G', Gil are virtual complex Lie subgroups of a Lie group G and let H, H' and H" be real forms of the groups G, G' and Gil, with H connected and H :J H', H" . Then H = H'H". Example 1. Consider the standard action of the group GL 2(1R) on ]Rpl. Since the subgroup O 2 :J GL 2(1R) is transitive on ]Rpl, we have the factorization

where B is the subgroup of all upper triangular matrices in GL2(1R). The group GL 2(1R) and its subgroups O 2, B are real forms of the group GL 2(C) and its algebraic subgroups 02(C) and B(C) (the subgroup of all upper triangular matrices). However, the group 02(C) is not transitive on !Cpl, so that GL 2(C) =f. 02B(C). Thus the converse to Lemma 4.7 is not valid. From this we can see that the factorization of Lie algebras g(2(C) = 02(C) + b(C), obtained from the global factorization g(2(1R) = 02 + b by complexification, is not itself global.

II. Lie Transformation Groups

143

In the classification of factorizations we make use of topological methods. Their application is based on the following theorem (see Onishchik 1962, Onishchik 1969). Theorem 4.4. Let G be connected Lie group, G', Gil two connected virtual Lie subgroups, such that the subgroup H = G' n Gil has a finite number of connected components. If G = G' Gil then there is an isomorphism of graded vector spaces E(G) E6 E(Ho) ~ E(G') E6 E(G").

(6)

If G is compact the converse also holds. Proof. The proof of equality (6) follows from Corollary 1 of Theorem 2.3 applied to the simply connected homogeneous covering space of the homogeneous space G' x Gil / H d , which is diffeomorphic to G. The proof of the converse uses the equality dimG = (2ml + 1) ... (2mr + 1), where ml, ... , mr are the exponents of the connected Lie group G. 0

The condition that H has a finite number of components in Theorem 4.4 is satisfied, in particular, in the following cases: G is an algebraic group (real or complex), Gil an algebraic subgroup; 7rl (G) is finite. Without this condition the theorem is false. Corollary 1. Let (G, G', Gil) be a factorization satisfying the conditions of the theorem. If K, K' and K" are maximal compact subgroups of G, G' and Gil such that K' c K, then K = K' K" and K' n K" is a maximal compact subgroup of the group H = G' n Gil . Corollary 2. Let (G, G', Gil) be a factorization satisfying the conditions of the theorem. Then either G' or Gil is a subgroup of maximal exponent in G. 4.5. Factorizations of Compact Lie Groups. In this subsection we present the classification of factorizations of simple compact Lie groups obtained in (Onishchik 1962). This work also provides a description of factorizations of arbitrary compact Lie groups, which we shall omit due to its complexity. Besides this, we shall give a description offactorizations (G, G', Gil) in which the subgroup G' n Gil is discrete. In the class of all factorizations of compact connected Lie groups we consider the equivalence relation generated by the local isomorphism and by permuting factors. The factorization (G, G, {e}) will be called trivial. In order to simplify the table we shall consider only irreducible factorizations (G, G', Gil), i.e. we shall assume that the action of G' x Gil by two sided translations on G is irreducible. This means that none of the subgroups G', Gil can be replaced in the factorization by its proper connected normal Lie subgroup. Taking into account these remarks we have Theorem 4.5 (Onishchik 1962). Every non-trivial irreducible factorization of a connected simple compact Lie group as a product of connected virtual

144

V. V. Gorbatsevich, A. L. Onishchik

Lie subgroups is equivalent to one of the factorizations (G, G', Gil) listed in Table 2 (all the subgroups in the table are considered as subgroups of G in the standard way.) Table 2

G SU2n

(n 2:

2)

S07

G'

G"

G'nG"

SP n

SU2n~1

SPn~l

S06

SU3

S05

SU2

G2

S02n

(n 2: 4)

S02n~1

SUn

SUn~l

G

G'

G"

G'nG"

S04n 2)

S04n~1

SP n

SPn~l

S016

S015

Sping

Spin7

S07

G2

S06

SU3

S05

SU2

(n 2:

SOS

Spin7

Proof. The proof of Theorem 4.5 is obtained according to the following scheme. Applying Corollary 2 of Theorem 4.4, we see that G' is a subgroup of maximal exponent in G. Hence G' is one of the subgroups listed in Theorem 2.2. Making use of Theorem 4.3 and the theory of linear representations of semi-simple Lie groups, we can determine all possible subgroups Gil (it turns out that the cases G = S08, G' = G 2 and G = E 6, G' = F 4 cannot be realized) . 0 Corollary 1. For any non-trivial irreducible factorization (G, G', Gil) of a connected simple compact Lie group G as product of virtual Lie subgroups G', Gil, the subgroups G', Gil, G'nG" are compact, simple and non-commutative, with G' = Nc(G')o.

In order to obtain all factorizations from Table 2, it suffices to compute the subgroup Nc(G")O for all subgroups Gil in this table. It is easy to see that NC(G")O = G"G~, where either G~ = {e} or G~ or G~ is a connected compact normal subgroup of rank l. Corollary 2. In any factorization of a simple compact Lie group G as product of connected virtual Lie subgroups G', Gil, both subgroups G' and Gil are compact, one of them is simple while the other is either simple or is a locally prime product of two simple subgroups, one of which has rank 1.

Note that almost all factorizations in Table 2 are connected with inclusions between transitive actions of simple compact Lie groups on spheres. Indeed, the series of factorizations of S02n, SU 2n , S04n coincides with the series of factorizations obtained in 4.4, and the factorizations S07 show that

= G 2 . S06,

S016

= S015 . Sping ,

S08

= Spin7 . S07

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Besides this, we see from the table that

St~,2 = G2/SU2,

St:,2

= SpindSU 3 ,

St:,3

= Spin7 /SU 2 .

Next we shall consider a result obtained (in the semi-simple case) in (Sternheimer 1968), which can also be deduced from the general classification of factorizations (see Onishchik 1969). We formulate it in the language of Lie algebras in the form proposed in (Koszul 1978). Theorem 4.6. Let (g, g', g") be a factorization of a compact Lie algebra such that g' n g" = O. Then a) There exists a direct sum factorization 9 = a' EEl a" such that the corresponding projections 9 ----) a' and 9 ----) a" define isomorphisms of g' onto a' and of g" onto a". In particular, 9 ~ g' EEl g" . b) The ideals a' and a" may be chosen in such a way that 3(a') = 7r(3(9')), 3(a") = 7r(3(9")), where 7r : 9 ----) 3(9) is the projection relative to the factorization 9 = 3(9) EEl [g, gJ. 4.6. Compact Enlargements of Transitive Actions of Simple Lie Groups. Here we shall consider actions of connected compact Lie groups which are proper extensions of transitive actions of connected simple compact Lie groups. As was shown in 4.2, to describe such enlargements it suffices to assume that they are irreducible. To begin with we shall display two classes of irreducible enlargements, which can be easily determined with the help of Theorem 4.4. Let (C', C, H') be a non-trivial factorization of a connected simple compact Lie group, where C, H' are Lie subgroups, C is connected and simple, H = G n H'. By Lemma 4.1, G/ H = G' / H'. Thus we have a proper irreducible enlargement of the action of G on G I H to an action of G'. An enlargement of this sort is said to be of type I. Embed a connected simple non-abelian compact Lie group G into C x G by the diagonal mapping g f---+ (g,g). Let (C,A',A") be a non-trivial factorization, H = A' n A". Then GI H = (G x C)/(A' x A") = GIA' x CIA" (see Corollary 4 of Lemma 4.1). The corresponding enlargement (of C to C x G) is said to be of type II. Consider an enlargement of type II. It turns out that the actions of C on CIA' and CIA" admit no enlargement of type II. Further, only one of these actions may admit an enlargement of type I. If such an enlargement C c C' exists, we obtain an enlargement of C to C' x C. Such a composition of enlargements of types II and I is said to be an enlargement of type III. Theorem 4.7 (Onishchik 1962). Any proper enlargement of a transitive action of a connected simple compact Lie group, which is an irreducible effective action of a connected compact Lie group, is an enlargement of type I, II or III.

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A homogeneous space M of a connected simple compact Lie group G is called exceptional if the action of G on M admits an enlargement of type I or II. Using Table 2 it is easy to enumerate all the exceptional homogeneous spaces. Corollary. Let M be a non-exceptional homogeneous space of a connected simple compact Lie group G and suppose the action of G on M is asystatic. Then this action does not admit any proper compact connected enlargement. The exceptional asystatic homogeneous spaces are (up to local similarity) the following ones:

S02n-t!Un- 1 , S04n-t!SPn-1 SPll S04n-t!SPn_1 U b S04n/SPn_1 SP1 SPn/SPn-1 U1 = cp2n-1, Spin7/G 2 = S7, Sping /Spin7 = S15, S015/Spin7' S016/Spin7' Spins/G 2 = S7 x S7, G 2 /SU 3 = S6. 4.7. Groups of Isometries of Riemannian Homogeneous Spaces of Simple Compact Lie Groups. Let M be a homogeneous space of a compact connected Lie group G, which acts on M effectively. By Corollary 1 of Theorem 3.8 there exist on M Riemannian structures invariant under the action of G. For any such structure a, the corresponding group of isometries Aut (M, a) is a Lie transformation group of M (see Chap. 1, Corollary of Theorem 2.11). This group is compact (see 3.4). Thus, we have a compact enlargement of the action of G on M. Using Theorem 4.2, its corollaries, Theorem 4.7 and its corollary we obtain the following statements. Theorem 4.8. Let a be an invariant Riemannian structure on a homogeneous space M = G / H, where G is a simple compact connected Lie group. If G / H is non-exceptional, then

Aut (M, a)o = GC

(locally direct product),

where C c (AutcM)o. If we exclude the case when M invariant then Aut (M,a) C SimcM and

= G and a is bi-

Aut (M, a) = GB,

where B

c

A

~

Aut (G,H).

Corollary 1. If, in addition, the action of G on M is asystatic, then

Aut (M,a)O

= G,Aut(M,a) = GB,

where B cAe Aut (G, H). Corollary of Lemma 4.2 implies Corollary 2 (see D'Atri and Ziller 1979). Let a be a left-invariant (but not bi-invariant) Riemannian structure on a connected simple compact Lie group G. Then G C Aut (M,a) C HolG i. e., Aut (M,a) = I(G)B,B c AutG.

II. Lie Transformation Groups

147

Now we apply these results to determine the group of isometries of the so called natural Riemannian structure. Let H be a Lie subgroup of a connected simple compact Lie group G and let beg be their tangent algebras. Consider the inner product on 9 invariant with respect to Ad G (which is unique up to a positive factor). Then 9 = bEBm where m = bl... Restricting the inner product to m we obtain, by Corollary 1 of Theorem 3.8, an invariant Riemannian structure 0"0 on M = G / H. It is called the natural Riemannian structure. It follows from Lemma 4.2 that 0"0 is invariant under SimcM. Using Theorem 4.8 and investigating thoroughly the exceptional homogeneous spaces, we come to the following result. Theorem 4.9. Let G be connected, simple and compact, M = G / H simply connected and 0"0 the natural Riemannian structure on M. Then Aut (M, 0"0)0

=

G(AutcM)o

locally direct product

Aut (M, 0"0) = SimcM = G· Aut (G, H), except for the following cases:

a) M = S6 = G 2 /SU 3 ,

Aut (S6, 0"0) = 0 7 ; b) M = S7 = SpindG 2 , Aut (S7, 0"0) = Os c) M = S7 X 8 7 = SpindG 2 Aut (S7 x S7, 0"0) = (Os x Os) ) O. If a connected semi-simple Lie group G has infinite centre, then the maximal compact subalgebra e of its tangent algebra 9 is not semi-simple. The description of the connected uniform subgroups of G is closely connected with the question: which uniform subalgebras ~ c 9 satisfy the condition 9 = ~ + fe, e]? For simple groups G this question is considered in (Gorbatsevich 1974), where, in particular, all uniform subalgebras ~ satisfying the above condition and such that ~ n fe, e] = 0 are found. A proof is given of the following Theorem 1.6. Let G be a connected Lie group and Ko its maximal compact subgroup. Then in any connected uniform Lie subgroup U of G there exists a connected Lie subgroup H C U, such that G = KoH and Ko n H = {e}. 1.5. Reductions of Transitive Actions of Reductive Lie Groups. Let G be a connected reductive Lie group, which acts transitively on a compact manifold G / H. In this subsection we will be concerned with describing virtual Lie subgroups G' c G, which act transitively on M. Consider a more general situation, when G = G' H, where G' and H are virtual Lie subgroups, and H is uniform. A subgroup A eGis called compact in G if the corresponding subgroup Ad A c GL(g) has a compact closure. The following theorem was proved in (Onishchik 1977). Theorem 1.7. Let (G, G' ,H) be a factorization, where G is a connected reductive Lie group, G', H its virtual Lie subgroups, with H uniform and not containing non-compact connected normal subgroups of G. Then the group G' is reductive and its radical is compact in G. If G is simple and non abelian and G'o does not contain connected normal subgroups G 1 -I G'o such that G = G1H, then G' is also simple and non abelian. Corollary 1. Let M = G / H be a compact homogeneous space, where G is reductive and acts on M locally effectively. If a virtual Lie subgroup G' C G acts transitively on M, then G' is reductive and its radical is compact in G. If G is simple and non abelian and G' acts transitively and irreducibly on M, then G' is simple and non abelian. Corollary 2. Let (G, G', Gil) be a factorization of a simple Lie group G as the product of virtual Lie subgroups G' and Gil. Then either G' or Gil is reductive in G and its radical is compact in G.

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Proof. If, for example, Rad Gil is non compact in G, then Gil is contained in a parabolic subgroup P of the group G. From Theorem 1.7 it follows that RadG' is compact in G. D A factorization (G, G', Gil) of a Lie group G is called maximal, if G' and Gil are maximal connected Lie subgroups of G. For a semi-simple group G, a maximal connected subgroup is either reductive (and its radical is compact in G), or it is parabolic. The classification of all maximal decompositions of non-compact simple Lie groups is given in (Onishchik 1969). The works (Nazaryan 1975a), (Nazaryan 1975b), (Nazaryan 1981) are devoted to the classification of arbitrary factorizations of these groups.

§2. Thansitive Actions on Compact Homogeneous Spaces with Finite Fundamental Groups Compact homogeneous spaces with finite fundamental groups (in particular simply connected ones) make up the most well researched class of compact homogeneous spaces. In this section we shall describe results on classification of transitive actions of Lie groups, and also on inclusions between transitive actions. Let M = G / H be a compact homogeneous space, with G connected and 71'1 (M) finite. Let us denote by S a Levi subgroup of G, by Ko a maximal compact subgroup of S and by K a maximal compact subgroup of G containing Ko. According to Corollary 4 of Theorem 3.11 of Chapter 2, the subgroups (K, K) and S act transitively on M. Hence the subgroup Ko also acts transitively. Therefore, the problem of describing of transitive actions on M falls naturally into the following stages: a) Describing transitive actions of connected compact (or connected compact semi-simple) Lie groups on M. b) For any connected semi-simple Lie group S, describing enlargements of transitive actions of a maximal compact subgroup Ko on M to actions of S. c) For any connected semi-simple Lie group S, describing enlargements of its transitive actions on M to actions of connected Lie groups, containing S as a Levi subgroup (enlargements of this type we shall call radical). Moreover, all these actions can be assumed to be locally effective. It turns out that solutions to problems b) and c) can be obtained in the general case. A much more difficult problem, whose solution is known only for certain classes of manifolds, is presented by a). Here we shall give its solution for simply connected compact manifolds of rank 1, which will enable us to give a description of transitive actions of Lie groups on spheres. The basic results on transitive actions of compact Lie groups on manifolds of other classes can be found in (Onishchik 1963), (Hsiang and Su 1968), (Onishchik 1970), (Scheerer 1971), (Schneider 1973), (Schneider 1975), (Hsiang 1975), (Mkhitaryan 1981), (Shchetinin 1988), (Shchetinin 1990) (see also the sur-

II. Lie Transformation Groups

179

veys (Alekseevskij 1974), (Alekseevskij 1979), (Vinberg 1963)). A survey of these results will be given in one of the following volumes of the present edition. We note that the first work devoted to this subject was the work of Montgomery and Samelson (Montgomery and Samelson 1943) on transitive actions of compact Lie groups on spheres, which to a substantial extent determined the direction and methods of future research. 2.1. Three Lemmas on Transitive Actions. In this subsection we shall give three general lemmas on compact homogeneous spaces G / H, where G is connected and H has a finite number of connected components. (the latter condition is satisfied if the group 7r1 (G / H) is finite). What these lemmas have in common is that both allow us to consider properties of transitive actions in terms of the topological structure of the manifold G / H. We note that by Theorem 1.2 of Chap. 2 the group of automorphisms of an arbitrary compact homogeneous space is a compact Lie group. Lemma 2.1 (see Onishchik 1966). Let M = G / H be a compact homogeneous space of a connected Lie group G, with H having a finite number of connected components. Then

rk AutcM ~ h( G / HO), where h is the homotopy characteristic (see 2.4 of Chap. 2).

For any reductive Lie group G let us denote by l (G) the number of its simple factors. Lemma 2.2 (see Onishchik 1968). Let M = G/H be a compact homogeneous space of a connected reductive Lie group G, with H having a finite number of connected components. If G acts irreducibly on M, then l(G) ~ rk G / HO. In the general case we have a locally direct decomposition G = GOG 1, where Go acts transitively and irreducibly, so that l(G o ) ::; rkG/Ho and G 1 is a Lie group with a compact tangent algebra, with rk G 1 ~ h( G / HO). In particular l(G) ~ 2rkG/Ho

In the following lemma we denote by rk Z (G) the rank of the centre of G (viewed as a finitely generated abelian group). Lemma 2.3 (OnisCik 1988). Let M = G/H be a compact homogeneous space of a connected semi-simple Lie group G, where G acts effectively on M and H has a finite number of connected components. Then

rkZ(G) ~ max{O,h(G/HO) -I}.

(4)

Corollary. If under the assumptions of Lemma 2.3 h( G / HO) ~ 1, then G has a finite centre. Example 1. Let M = SUn, n 2: 3. Then h(M) = n -1. Using the results of (Gorbatsevich 1974) (see Theorem 1.6 above) one can construct an effective action on M of the universal covering G = SU1,n of the group SU1,n' The

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V. V. Gorbatsevich, A. L. Onishchik

group G has an infinite centre of rank 1. In particular, the bound (4) attained in the case n = 3.

IS

2.2. Radical Enlargements. This subsection is devoted to problem c) formulated at the beginning of this section. Let again M = G / H be a compact homogeneous space with a finite fundamental group, with G connected and acting on M locally effectively. Let T be the largest connected triangular subgroup of the group Rad G, which by Corollary 2 of Theorem 1.1 is contained in Nc(HO), and in view of Corollary 5 of the same theorem is isomorphic to ]RP and coincides with the nilpotent radical of the group G. The tangent algebra of the group G can be naturally identified with a subalgebra 9 C tJ(M) and to the subgroup T corresponds a nilpotent radical t C 9. If the subgroup H is connected, then T C Nc(H) and T acts on M by automorphisms of the homogeneous space (see formula (1) of Chapter 2). To this action corresponds a commutative subalgebra a C u(M)). Lemma 2.4. Suppose that H is connected. Then t C CPa, where cP is some finite-dimensional G-submodule in

F(M)U = {¢ E F(M) I v¢ = 0

for all v E a}.

In particular, t is a compact G-module. Proof. The second statement of the lemma follows from Lemma 1.2 of Chap. 1. D From this lemma one can deduce the following theorem, which shows that our assumptions impose fairly strong restrictions on the structure of G. Theorem 2.1 (Vishik 1973, see also Onishchik 1977). Let G be a connected Lie group, which transitively and locally effectively acts on a compact manifold M = G / H, where 7fl (M) is finite. Then

G = G1

~

N,

where G 1 is a connected reductive Lie subgroup, N a vector Lie group, the action of G 1 on N by inner automorphisms is compact and NCI = O. The tangent algebra of the subgroup N coincides with [9, rad 9J· Proof. By passing to a finite covering G / HO of the manifold M, we can assume that the group H is connected. From Lemma 2.4 it follows that the nilpotent radical of the Lie algebra 9 is a compact G-module. Let s be a Levi subalgebra of the Lie algebra 9. One can show that the s-invariant complement c to the ideal n = [9, rad 9J in rad 9 is an abelian sub algebra. Putting 91 = s EB c, we obtain a semi-direct factorization 9 = 91 e- n. The corresponding decomposition of G is the one we are seeking. D Corollary 1. Suppose that under the assumptions of Theorem 2.1 G = SnSeC, where C = Rad G 1 and Se (correspondingly Sn) is the product of all compact (non-compact) simple connected factors of the group (G 1 ,Gd. Then the decomposition G = SnGO, where Go = (SeC) ~ N, is locally direct.

II. Lie Transformation Groups

181

Corollary 2. In the notation of Corollary 1 we have

Corollary 3. If the subgroup Bn C G acts transitively on M then the group G is reductive. Let us now consider how this result can be applied to the solution of problem c). Suppose that M = B/F is a compact homogeneous space of a connected semi-simple Lie group B, acting locally effectively on M, 1Tl(M) is finite and suppose that we have a locally effective action of a group G with B as a Levi subgroup, which extends the given action of B. The structure of the group G is described by Theorem 2.1, and moreover, one can assume that B C G1 • The enlargement B C G1 can be constructed easily with the help of 4.3 of Chap. 2 (it it a reduction of the natural enlargement of the group B). The enlargement G 1 C G can be described with the help of Lemma 2.1 (here one has to pass to a finite covering of the manifold M). Conversely, if a is any commutative subalgebra in tl(M)G" then F(M)na is a G1-invariant commutative sub algebra in tl(M) Any finite-dimensional G1-submodule n C F(M)na, such that n G1 = 0, defines an action of the Lie group G = G 1 ~ n on M, which is an enlargement of the given action of the group G 1 • For the description of radical enlargements in terms of stabilizers see (Onishchik 1966), (Vishik 1973). Let B = BnBe be a locally direct factorization, where Be is the largest connected compact normal Lie subgroup in Band Bn a connected normal Lie subgroup without compact factors. According to Corollary 3 of Theorem 2.1, if the subgroup Bn is transitive on M then the group G = G 1 is reductive. In the contrary case, making use of Corollary 3 and Theorem 2.1 of Chap. 3 we obtain

Corollary 4. If M = B/ F, the subgroup Bn does not act transitively on M and the action of B on M is asystatic, then there exists a radical enlargement of the action of the group B on M with a radical of arbitrarily large dimension. 2.3. A Sufficient Condition for the Radical to be Abelian Theorem 2.2 (Onishchik 1966). Let M = K/L where K is a connected semi-simple Lie group, and let rkK AutK K / LO :S 1. If the action of K on M is a reduction of a locally effective action of some Lie group G :J K, then Rad G is abelian. Note that in the case when rkAutKK/Lo = 0, i.e. when NK(LO)O = LO, the group G is semi-simple (see Corollary 3 of Theorem 1.1). Using Lemma 2.1 we obtain

Sf

Corollary. Let M be a compact manifold with a finite fundamental group, its universal covering space, G a Lie group which acts transitively and

182

V. V. Gorbatsevich, A. L. Onishchik

locally effectively on M. If h(M) = 0 then G is semi-simple and if h(M) = 1 then Rad G is abelian. Example 2 (Onishchik 1966). Let M = S2n-l, n 2: 2. Since h(M) = 1, every Lie group acting transitively on M has an abelian radical. Such, in particular, are the radical enlargements of the standard action of SUn on s2n-l. As we saw in 1.1 of Chap.2 (Example 1.1), this action is systatic, with AutsunS2n-l consisting of transformations )"'E,)... E C X (using the standard inclusion s2n-l C cn), and its tangent algebra is the one-dimensional subalgebra a = CVo, where vo(z) = iz (z E S2n-l). The algebra F(s2n-l)o is naturally isomorphic to the function algebra F(cpn-l). For any finitedimensional SUn-submodule C F(cpn-l) the space r = a = vo is a commutative sub algebra in l1(S2n-l) invariant under SUn. We obtain a locally effective action of the group SUn ~ r with abelian radical r on s2n-l. By Lemma 2.1 any radical enlargement of the action of SUn has this form. Here the radical r can have arbitrarily large dimension (cf. Corollary 4 of Theorem 2.1). The following example shows that the conditions imposed on M in the corollary to Theorem 2.2 are essential. Example 3 (Onishchik 1966). Let M = SU 3 . Then h(M) = 2. Let K = U 3 , L ~ U 1 . be the subgroup of matrices of the form diag (I,E, 1), where lEI = 1. Then K = SU3 · L, SU 3 n L = {e}, so that K/L = SU3 . Consider the group G = U 3 I>< C3 , where the representation of U3 in C3 is the standard one, and the subgroup H, consisting of pairs of the form

' (1 ,E,e 27riRez 1 ) , (dlag

(

Zl,Z2,Z3 ) ) ,

where lEI = 1, (Zl,Z2,Z3) E C3 . It is easy to see that G = KH, KnH = L and that H does not contain non-trivial normal subgroups of G. Thus there exists an effective action on SU 3 of the group G, which is an enlargement of the given action of K = U 3. Moreover, Rad G = Z (U 3) I>< C3 is non-abelian. 2.4. Passage from Compact Groups to Non-Compact Semi-simple Groups. In this subsection we consider problem b), for simplicity limiting ourselves only to the case of connected stabilizers. Let G be a connected semi-simple Lie group and K a connected Lie subgroup corresponding to a maximal compact subalgebra t of its tangent algebra g. A connected Lie subgroup L C K will be called a G-subgroup if the standard action of K on K / L extends to an action of G. These subgroups possess a simple description. Namely, let G = K P be a Cartan decomposition of G. For any point pEP denote by Lp the identity component of the subgroup

Let Mp be the largest connected normal Lie subgroup in Zc(p)

=

{g E G I gpg-l

= p},

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II. Lie Transformation Groups

contained in K. Then we have a locally direct decomposition Lp where Lp * is a connected normal Lie subgroup of Lp. Theorem 2.3 (Onishchik 1967). A subgroup L only if it has the form L

cK

=

MpLp *

is a G-subgroup if and

= MLp*,

where pEP and M is a connected Lie subgroup of Mp.

We can also describe (in terms of Lie algebras) all possible enlargements of the standard action of K on K / L to an action of the group G. Corollary 1. If the tangent algebra of G is split then G -subgroups in K are subgroups of the form L p, where pEP. Corollary 2. Let G = K(C) be a connected semi-simple Lie group. A subgroup L c K is a G-subgroup if and only ifD 1 Z K (k)0 C L c ZK(k)O, where ZK(k) = {g E K I gkg- 1 = k} for some element k E K.

This corollary gives rise to the description of simply connected compact complex homogeneous spaces obtained in (Wang 1954) (cf. Corollary 2 of Theorem 1.5). Corollary 3. Suppose that a G-subgroup L does not contain connected normal subgroups of the group K and is maximal among connected Lie subgroups of K possessing this property. Then L = L p, for some pEP.

We note that for pEP, the subgroup ZK(p) can be interpreted as the stabilizer of the point p with respect to the action (k, x) f-+ kxk- 1 (k E K, x E P), of K on P by inner automorphisms. Thus K / Lp = K / Z K (p)O is a homogeneous space, which covers the orbit K(p) C P. Instead of K(p) one can consider the homogeneous space (Ad) K(s) isomorphic to it, where 8 E p, exp s = p and 9 = t EB P is a Cartan decomposition of the Lie algebra 9 of G. Example 4. Suppose that the real rank of G is equal to 1, i.e. that dim II = 1 in the notation of 1.3. In this case the group K acts transitively (by means of the adjoint representation) on the set of all one-dimensional subspaces of p (Helgason 1962). Hence K acts transitively on the sphere SN-1 = {s E pi (8, s) = I}, where N = dimp, and ( , ) is the Killing form on g. Making use of the classification of simple groups of real rank 1 we obtain transitive actions of the following non-compact simple groups on spheres: 1) G = SO~,n on sn-l (K = SOn), 2) G = SU 1,n on s2n-1 (K = Un), 3) G = SP1 n on s4n-l (K = SPn x SPn), 4) G = FU'on S15 (K = Sping).

Moreover, the action of K extends to an action of G in a unique way; the stabilizer H c G coincides with the unique (up to conjugation) proper connected parabolic subgroup of G.

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V. V. Gorbatsevich, A. L. Onishchik

Let us give the classical geometric description of the action of the group 01,n on sn-l, of which the above action of SO~,n is the reduction. The group + xi + ... + = 0 in lRn+1 , which leads 01,n naturally acts on the cone to its transitive action on the quadric Q C IRP n , given by the same equation in homogeneous coordinates. Clearly, Q is contained in an affine open set (xo =I 0) and in non-homogeneous coordinates ~i = :~ (i = 1, ... , n) is given by the equation ~r+" .+~~ = 1. Thus Q = sn-l. Analogously, (with the help of projective spaces over C or JHI) one constructs actions of U l,n and SPl,n on s2n-l and s4n-l respectively and (making use of the octal projective space) of the group FII on S15.

-x5

x;

Example 5. Let G~ k be the manifold of all oriented k-dimensional subspaces of the space lRn (the double covering of the Grassmann manifold G~,k)' The natural action of the group SLn(lR) on G~,k and its reduction to the

subgroup SOn are transitive, with G~,k = SOn/SOk x SOn-k. Clearly, the homogeneous SOn-spaces G~,1 and G~,n-l are isomorphic to the sphere sn-l with the standard action of the group SOn. At the same time, for n 2: 3 the actions of SLn(lR) on these manifolds are non-isomorphic (although they are similar). The corresponding stabilizers are two non-conjugate maximal connected parabolic subgroups of SLn(lR), which are taken into each other by an inner automorphism ofthis group. Thus on sn-l (n 2: 3) there exist two similar, but non-isomorphic actions ofthe group SLn(lR), which are enlargements of the standard action of the group SOn2.5. Compact Homogeneous Spaces of Rank 1. In this subsection we will give a solution to problem a) for simply connected compact homogeneous spaces of rank 1 (in the sense of 2.4 of Chap. 2). In view of Lemma 2.2 we can restrict ourselves to the case when the compact group which acts transitively, is simple and non-abelian. Therefore we shall start with listing (up to a local isomorphism) all pairs of connected compact Lie groups (G, H) such that HcGandrkG/H=1. Theorem 2.4 (Onishchik 1963). All connected Lie subgroups H of connected simple non-abelian compact Lie groups G, such that rk G / H = 1, are given in Table 3. The groups G are considered up to local isomorphism, and subgroups H up to conjugation in G.

In Table 3 we denote by is a finite normal subgroup of IT, then ITjlf> ~ IT. Both relations - commensurability and weak commensurability - are equivalence relations on the class of all groups. For torsion free groups commensurability and weak commensurability are, clearly, equivalent. Let Me ----; M' ----; Ma be the natural bundle for a compact homogeneous space M (here M' is some finite covering manifold of M). From the exact homotopy sequence of this fibre bundle it follows that 7fdMa) ~ 7fdM') ~ 7f1 (M). Thus, the group 7f1 (Ma) is determined by 7f1 (M) up to weak commensurability. Since 7f1 (Ma) is torsion free it is in fact determined by 7fd M) up to commensurability. Note that this statement is used in an essential way in the proof of Theorem 4.4.

II. Lie Transformation Groups

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5.2. Embedding of the Fundamental Group in a Lie Group. The following theorem (as well as Theorem 4.2) is the foundation on which one can build proofs of the basic results on the fundamental group of compact homogeneous manifolds. This allows one to reduce almost completely the study of the group 11"1 (M) to the study of lattices in Lie groups. Theorem 5.1 (Gorbatsevich 1979). Let M be a compact homogeneous manifold. Then in the group 1I"1(M) there exists a subgroup 11"' of finite index, isomorphic to a uniform lattice in some connected Lie group F. Proof. We give a sketch of the construction of F. Let G be a connected Lie group acting transitively on M. By Theorem 4.1 we shall suppose that the action of G on M is proper. If H is the stabilizer of this action then it turns out that one can take as F the group (NG(HD)D / HD X ]Rn for some n 2: 0 (uniquely determined by G and H). D

Note that for an arbitrary (not necessarily compact) homogeneous manifold M there exists a subgroup 11"' C 11"1 (M) of finite index, isomorphic to a discrete subgroup of a connected Lie group. An arbitrary connected Lie group is locally isomorphic to some linear Lie group. Hence by Theorem 5.1 the properties of the group 1I"1(M) should be rather close to properties of linear groups. For example, the Tits alternative holds for 11"1 (M) (for linear groups see Merzlyakov 1980): the group 11"1 (M) is either commensurable with a solvable group or it contains a free abelian subgroup. Another example is provided by Theorem 5.2 below. 5.3. Solvable and Semi-simple Components. Let II be an arbitrary group. The radical of II is the largest solvable normal subgroup of II (when one exists). Theorem 5.2 (Gorbatsevich 1981a). Let M be a compact homogeneous manifold. Then in 11"1 (M) there exists a radical. Proof. We give a sketch of the proof. In view of the compactness of M the group 11"1 (M) is finitely generated. Hence, if 11"1 (M) is isomorphic to a linear group, the existence of the radical is a classical result (Auslander 1973). In the general case from Theorem 5.1 one can easily deduce that 1I"1(M) is commensurable with a group with a radical. But then one can show that 11"1 (M) also must have a radical. . D

Consider now the structure bundle Mr -+ Ma -+ Ms for a compact homogeneous manifold M. From the homotopy exact sequence of this fibre bundle we obtain the sequence

from where it is clear that the group 11"1 (Ma) can be viewed as an extension of the group 11"1 (Mr) by 11"1 (Ms). From Borel's density theorem for lattices in semi-simple Lie groups (see Part I of Encycl. Math. Sc. 21) it follows that the

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radical of the group 7fd Ms) is trivial. Therefore 7fd Mr) is isomorphic to the group r( 7f1(Ma)), and hence 7f(Ms) C::' 7f1(Ma)/r(7f1(Ma)). Thus, the groups 7fdMr) and 7f1(Ms) are determined by the group 7fdMa) uniquely up to isomorphism. They are also determined by the group 7f( M) up to commensurability. The group 7f1 (Mr) is called the solvable part and the group 7f1 (Ms) the the semi-simple part of the group 7f1 (M). They determine the smooth manifolds Mr and Ms uniquely up to a finite covering (see Theorem 4.4). In some cases exact sequence (*) splits (with a suitable choice of the base Ma of the natural bundle), i.e. the group 7f1 (Ma) is a semi-direct product of 7f1(Ms) and 7f1(Mr)). This is the case, for example, when the Lie group G is complex. But in general sequence (*) does not always split, even after passage to a finite covering of M (or Ma). For example, consider M = SL 2 (IR.)/r, where r is a uniform torsion free lattice. Then exact sequence (*) has the form

(see Example 2 in 4.3). We see that 7fdM) is an extension of the group Z by the group 7f1 (Fg), and it is known that the characteristic class of this extension is non-trivial (Auslander et al. 1963). Hence, in the case we are considering, the exact sequence (*) does not split.

5.4. Cohomological Dimension. The cohomological dimension of a group 7f1 (M) is an important invariant. To study it we need certain auxiliary notions, which, in slightly lesser generality, were considered by Serre (1971). A group II is called virtually torsion free, if it is weakly commensurable with a torsion free group (in Serre's definition commensurability is required). If II is a finitely generated linear group, then it contains a torsion free subgroup of finite index, i.e. II is virtually torsion free in the sense of Serre (see Part I of Encycl. Math. Sc. 21). If, however, II is a finitely generated subgroup of a connected Lie group, then II is not necessarily virtually torsion free in the sense of Serre, but it is such in our sense. Moreover, if M is a compact homogeneous manifold, then 7f1 (M) ;:::0 III (Ma), so that 7f1 (M) is virtually torsion free. A group II is called a group of type (FL), if the group Z, viewed as a trivial II module, has a finite free resolution. For such groups one defines the cohomological dimension cd to be the minimal length of a resolution of the above kind. All groups of type (FL) are torsion free. Hence it is useful to introduce the notion of virtual cohomological dimension vcd II (also slightly more general than its namesake due to Serre (1971)), which is finite even for some torsion groups. We shall call II a group of type (VFL) (Le. virtual (FL) type), if it is weakly commensurable with some group II' of type (FL). For such a group II we put vcd II = cd II' (one can show that vcd II is well defined). Properties of the invariant vcd II are close to those given in (Serre 1971) in a slightly less general situation.

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If II is a polycyclic group, then vcd II is equal to the rank (or the Hirsch number) of the group II, for whose definition see Part I Encycl. Math. Sc. 21 or (Auslander 1973). The role ofvirtual cohomological dimension in the study of compact homogeneous manifolds, can be seen, for example, from the following result. Theorem 5.3. Let M be a compact homogeneous manifold. Then (i) The group 7I"1(M) is of type (VFL). (ii) vcd7l"1(M) = vcd7l"1(Ma ) = dimMa and vcd71"1 (M) ~ dimM. (iii) vcd 71"1 (M) = dim M if and only if, when M is aspherical. In the contrary case vcd 71"1 (M) ~ dim M - 2.

Note that by statement (ii) the number dimMa depends only on the group 71"1 (M). The group 71"1 (Ma) (and also 71"1 (M)) is torsion free and is clearly a Poincare Duality group (Le. a PD-group, see Part I of Encycl. Math. Sc. 21). 5.5. The Euler Characteristic. If II is a group of type (FL), then its Euler characteristic is defined by the formula 00

X(II) = ~) _1)ibi , i==O

where bi = dim Hi (II, Q) is the i-th Betti number. Suppose now the group II satisfies the weak (VFL) condition. By definition, II contains a subgroup II' of finite index, which has a finite normal subgroup such that II' / is a group oftype (FL). We define the Euler characteristic by the formula

X(II) =

I~~~'I X(II' /2) be the total space ofthe (unique) non trivial line bundle over ru>2, and S(ru>2) the total space of the (unique) non trivial fibre bundle over ru>2 with fibre S1 and structure group ~. By L(K2) we denote the total space of the (unique) non-trivial line bundle over K2. These three manifolds are homogeneous (Gorbatsevich 1977a)j they are homogeneously indecomposable. Theorem 2.3 (Gorbatsevich 1977a). Any homogeneous, homogeneously indecomposable three-dimensional manifold is diffeomorphic to one of the following: (i) A solvmanifold: L(K2) or a compact one (in particular, N s (lR)/N 3 (Z), lR ~ lR2 IZ ~'" Z2 - see Example 5c). (ii) L(ru>2), S(ru>2).

(iii) SU 2 ID,

where D is a finite subgroup of SU2

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(iv)

Air

where

r

is a non abelian discrete subgroup of A.

The manifolds mentioned in various parts of Theorem 2.3 are not diffeomorphic to one another. Two solvmanifolds in (i) are diffeomorphic if and only if their fundamental groups are isomorphic. The manifolds in (ii) are mutually non diffeomorphic and the manifolds in (iii) are diffeomorphic if and only if the groups D are isomorphic. We note that three-dimensional homogeneous manifolds are homeomorphic if and only if they are diffeomorphic. From the above results one can, in particular, deduce that a compact three-dimensional manifold M is determined up to diffeomorphism by its fundamental group. For non compact manifolds this is not always the case: ]Rl x S2 and ]R3 are homogeneous and simply connected, however they are not diffeomorphic. The statement is also false for four-dimensional simply connected homogeneous manifolds (for example for S2 x S2 and S4). The aim of describing arbitrary transitive actions on manifolds M for dim M :::: 3 is at present not attainable. Therefore from among all transitive actions it is convenient to separate and study those which are minimal in the sense defined below. A transitive action of a connected Lie group G on a manifold M is called minimal, if it is locally effective and if G does not contain subgroups acting transitively on M. Finding all transitive actions of Lie groups on a given manifold M reduces to a) Finding of all minimal actions on M. b) Finding of all extensions of each of the minimal actions found in a). Taking (Mostow 1950) as the starting point, it is possible to compute all minimal transitive actions of Lie groups on surfaces. All minimal actions on three-dimensional manifolds are described in (Gorbatsevich 1977a).

§3. Compact Homogeneous Manifolds of Low Dimension 3.1. On Four-dimensional Compact Homogeneous Manifolds. In §2 we considered homogeneous manifold of dimension ::; 3. As the dimension increases the number of homogeneous manifolds, even just the compact ones, grows and their description becomes considerably more complicated. At the present time (in 1992) there are described up to diffeomorphism compact homogeneous manifolds only up to dimension 4 (for partial results in higher dimensions see below). Homogeneously indecomposable compact homogeneous four-dimensional manifolds are the solvmanifolds (the number of which is even), 7 separate manifolds and 4 infinite series, for details see (Gorbatsevich 1977b). Let us consider some concrete examples. The manifolds S4, JlU>4 and CP2 are homogeneous and homogeneously indecomposable. Also homogeneous are the total spaces of bundles with fibre

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S1 over a base of the form SU 2/ D, where D is a finite subgroup of SU 2. All these spaces admit a finite covering by the manifold S1 x S3. We note that on some compact four-dimensional manifolds M there exists (up to isomorphism) only one minimal transitive action of a Lie group - for example for M = S4 this is the natural action of the group S05. Some M admit even minimal actions of Lie groups of arbitrarily large dimension (for example the torus 1[4). 3.2. Compact Homogeneous Manifolds of Dimension ::::: 6. For dim M ~ 5 determination of compact homogeneous, homogeneously indecomposable manifolds M has not so far been achieved. However, for dim M ::::: 6 one can describe such manifolds up to a finite covering. Theorem 3.1 (Gorbatsevich 1980). Let M be a compact homogeneous manifold of dimension ::::: 6 Then there exists a manifold M', which is a finite covering of M, and which decomposes as a direct product of manifolds from the following list {in which all manifolds are homogeneous and homogeneously indecomposable} : {i} Simply connected manifolds: Sk {2::::: k ::::: 6}, Cpk {k = 2, 3}, SU3/S0 3, SU 3/1'2 {where 1'2 is a maximal torus in SU3}, G~2 {Grassmann manifold}. {ii} Solvmanifolds of the form R/r where r is' a lattice in some simply connected solvable Lie group R, with dim R = 1,3,4,5 or 6 and the group r does not decompose as a direct product of proper subgroups. {iii} A/r, A x A/ D, where A = SL 2 (lR), rand D are uniform lattices with D indecomposable. {iv} A ~ Ad lR3/r, where r is a uniform lattice in the Lie group A ~ Ad lR3 the semi-direct product of the groups A and lR3 , corresponding to the adjoint representation Ad : A ---) GL3(lR). {v} S3 x Fg, where Fg is a closed orientable surface of some genus g ~ 2. {vi} SL 2 (CC)/r, where r be a uniform lattice. Moreover, the direct factors of M', which appear above, are determined by the original manifold M uniquely up to a finite covering and the order of listing. The proof of this theorem is based on the general results on compact homogeneous manifolds given in Chap. 5. In particular, for a suitable M', which is a finite covering of M, one considers the natural bundle and the Borel bundle, and then makes use of topological methods of classifying such bundles. If the fibre and the base degenerate to a point, then one applies other, special methods. For example, if M is aspherical, then one makes use of results of (Gorbatsevich 1983b). If M is simply connected, then for dimM = 6 see (Gorbatsevich 1980) and for dimM < 6 the argument is simple. From Theorem 3.1 it follows, in particular, that for dim M ::::: 6 for a compact homogeneous manifold M there exists a finite covering M', such that the natural bundle for M' is trivial.

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Another corollary of Theorem 3.1 concerns the determination of a compact homogeneous manifold M by its homotopy groups, or, more generally, by its homotopy type. If Ml and M2 are compact manifolds and Ml is homotopy equivalent to M 2 , than sometimes it is possible to show that Ml is diffeomorphic to M 2 • For example, from Theorem 3.1 it easily follows that this is true if dim Mi ~ 6 and Mi are simply connected (i = 1,2). In the general case, homotopy equivalent compact homogeneous manifolds are not always diffeomorphic (for details see 2.5 of Chap. 5 and also 3.3 below). With the help of Theorem 3.1 one can prove the following statement. Theorem 3.2 (Gorbatsevich 1980). Let Ml and M2 be compact homogeneous manifolds of dimension ~ 6. Then, if the homotopy groups 7l"i(Md and 7l"i(M2) are isomorphic for 1 ~ i ~ 5, then Ml and M2 are diffeomorphic up to a finite covering.

Let us now consider the question of the relation of decomposability and homogeneous decomposability for compact manifolds of low dimension. One can prove that, if M is a compact homogeneous manifold of dimension ~ 4 and M is diffeomorphic to Ml x M2 (i.e. M is decomposable), then the manifolds Ml and M2 are homogeneous. (Hence, for compact homogeneous manifolds of dimension ~ 4 the properties of decomposability and homogeneous decomposability are equivalent.) In dimensions ;::: 5 this no longer holds. For example, the manifold S3 x Fg is homogeneous (see Theorem 3.1 (v)), but Fg for g ;::: 2 is not homogeneous (see, for example, Theorem 2.2). A few words concerning the complex case. Let M be a compact complex homogeneous manifold and dimcM ~ 3. Then dimIRM :S 6 and hence up to a finite covering the smooth type of the manifold M is described by Theorem 3.1. In fact, if dimcM = 1, then M is diffeomorphic to 1f2 or S2, and for dimcM = 2,3 the description of all such M is given in (Tits 1962). 3.3. On Compact Homogeneous Manifolds of Dimension ;::: 7. Attempts to classify compact homogeneous manifolds of dimension 7 and above (even up to finite coverings) meet with the following obstacles. First, there exist already an countably infinite number of mutually non diffeomorphic simply connected homogeneous manifolds M of dimension 7 (from Theorem 3.1 it follows that for dimM ~ 6 the number of such M is finite). For example, the set of mutually non diffeomorphic manifolds of the form SU3/1I', where 1I' is the one-dimensional torus in SU3 is countably infinite. Another countably infinite series consists of manifolds of the form SU3 x SU 2/ H (see 2.5 of Chap. 5), where for the stabilizer H = SU2 . T, the semi simple part L = SU2 is embedded in the standard way in SU3 and T is some one-dimensional torus, which normalizes the subgroup L, with L n T discrete. It turns out (Kreck and Stolz 1988), that among manifolds of this form there exist the following: a) Homotopy equivalent but non-diffeomorphic manifolds.

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b) Homeomorphic but not diffeomorphic manifolds (see Example 7 of 2.5 of Chap. 5). Further, beginning with dimension 7, there exist compact homogeneous manifolds M, such that for any finite covering M' of M, the natural bundle is non trivial (and is not diffeomorphic to Me X M a, which makes classification very much harder), for examples see 3.3 of Chap. 5. Finally, the classification of compact solvmanifolds M is very complicated (and at the present time has not been explicitly carried out) already in dimensions 5, 6 and for dimM = 7 the complications appear, at this time, to be insurmountable.

References* Akhiezer, D. N. (1974): Compact complex homogeneous spaces with solvable fundamental group. Izv. Akad. Nauk SSSR, Ser. Mat. 38, No.1, 59-80. English transl.: Math. USSR, Izv. 8, 61-83. Zbl.309. 22008 Alekseevskij, D. B. (1974): Lie groups and homogeneous spaces. Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 11,37-123. Zbl. 296. 22010. English transl.: J. SOy. Math. 4,483-539 Alekseevskij, D. B. (1979): On proper actions of Lie groups. Usp. Mat. Nauk 34, No.1, 219-220. Zbl. 422. 53020. English transl.: Russ. Math. Surv. 34. No.1, 215-216 Alekseevskij, D. B. (1982): Lie groups. Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 20, 153-192. Zbl. 532. 22010. English transl.: J. Sov. Math. 28, 924-949 (1985) Auslander, L. (1973): An exposition of the structure of solvmanifolds. I, II. Bull. Am. Math. Soc. 79, No.2, 227-261, 262-285. Zbl. 265. 22016 and Zbl. 265. 22017 Auslander, L., Green, L., Hahn, F. (1963): Flows on Homogeneous Spaces. Ann. Math. Stud., No. 53, Princeton Univ. Press, Princeton. Zbl. 106, 368 Auslander, L., Szezarba, R. (1962): Characteristic classes of compact solvmanifolds. Ann. Math., II. Ser., 76, No.1, 1-8. Zbl. 114,399 Auslander, L., Szczarba, R. (1975a): Vector bundles over tori and noncompact solvmanifolds. Am. J. Math. 97, No.1, 260-281. Zbl. 303. 22006 Auslander, L., Szczarba, R. (1975b): On free nilmanifolds and their associated non-compact solvmanifolds. Duke Math. J. 42, No.2, 357-369. Zbl. 333. 22005 Auslander, L., Tolimieri, R. (1975): Abelian harmonic analysis, theta functions and function algebras on a nilmanifold. Lect. Notes Math. 436, Springer, Berlin. Zbl. 321. 43012 Barth, W., Otte, M. (1969): Uber fast-uniforme Untergruppen komplexer Liegruppen und auflosbare komplexe Mannigfaltigkeiten. Comment. Math. Helv. 44, No.3, 269-281. Zbl. 172,378 Barut, A. 0., Rlj.czka, R. (1977): Theory of Group Representations and Applications. Polish Scient. Publishers, Warsaw Borel, A. (1950): Le plan projectif des octaves et les spheres comme espaces homogenes. C. R. Acad. Sci., Paris 230, No. 15, 1378-1380. Zbl. 41, 522

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Borel, A. (1953): Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts. Ann. Math., II. Ser. 57, No.1, 115-207. Zbl. 52, 400 Borel, A. (1960): Seminar on transformation groups. Ann. Math. Stud. No. 46, Princeton Univ. Press, Princeton. Zbl. 91, 372 Borel, A., Tits, J. (1965): Groupes reductifs. Publ. Math., Inst. Hautes Etud. Sci. 27, 659-755. Zbl. 145, 174 Bourbaki, N. (1960): Topologie generale, Ch.3, 4. 3-eme ed. Hermann. Paris. Zbl. 102,271 Bourbaki, N. (1968): Groupes et algebres de Lie. Ch.4-6. Hermann, Paris. Zbl. 186, 330 Bourbaki, N. (1982): Groupes et algebres de Lie, Ch.9. Masson, Paris. Zbl. 505. 22006 Bredon, G. E. (1972): Introduction To Compact Transformation Groups. Acad. Press, New York, London. Zbl. 246. 57017 Chebotarev, N. G. (1940): Theory of Lie Groups. GITTL, Moscow, Leningrad (Russian) Chevalley, C. (1946): Theory of Lie Groups. Vol. 1. Princeton Univ. Press, Princeton D'Atri, J. E., Ziller, W. (1979): Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups. Mem. Am. Math. Soc. 18, No.215. Zbl. 404. 53044 Dao Van Tra (1975): On spherical sections on a compact homogeneous space. Usp. Mat. Nauk 30, No.5, 203-204 (Russian). Zbl. 321. 57026 Dao Van Tra (1981): On extensions of groups acting transitively on compact manifolds. In Geom. Metody Zadachakh Algebry Anal., 87-106 (Russian). Zbl. 485. 22019 Dubrovin, B. A., Novikov, S. P., Fomenko, A. T. (1980): Modern Geometry. Nauka, Moscow. Zbl. 433. 53001. English transl.: Grad. Texts Math. 93, 1984, and 104, 1985. Springer, New York Dynkin, E. B., Onishchik, A. L. (1955): Compact global Lie groups. Usp. Mat. Nauk 10, No.4, 3-74. Zbl. 65, 262. English transl.: Amer. Math. Soc. Transl. 21 (1962), 119-192 Eisenhart, L. P. (1933): Continuous Groups of Transformations. Princeton University Press, Princeton; Humphrey Milford, London. Zbl. 8, 108 Fujimoto, H., (1968): On the holomorphic automorphism groups of complex spaces. Nagoya Math. J. 33, 85-106. Zbl. 165, 404 Golubitsky, M. (1972): Primitive actions and maximal subgroups of Lie groups. J. Differ. Geom. 7, No. 1-2, 175-191. Zbl. 265. 22024 Golubitsky, M., Rothschild, B. (1971): Primitive subalgebras of exceptional Lie algebras. Pac. J. Math. 39, No.2, 371-393. Zbl. 209, 66 Gorbatsevich, V. V. (1974): On a class of factorizations of semi-simple Lie groups and Lie algebras. Mat. Sb., Nov. Ser. 95, No.2, 294-304. Zbl. 311. 22017. English transl.: Math. USSR, Sb. 24, 287-297 Gorbatsevich, V. V. (1977a): On three-dimensional homogeneous spaces. Sib. Mat. Zh. 18, No.2, 280-293. Zbl. 373. 14017. English transl.: Sib. Math. J. 18, 200-210 Gorbatsevich, V. V. (1977b): On the classification of four-dimensional compact homogeneous spaces. Usp. Mat. Nauk 32, No.2, 207-208 (Russian). Zbl. 362. 57020 Gorbatsevich, V. V. (1977c): On Lie groups acting transitively on compact solvmanifolds. Izv. Akad. Nauk SSSR, Ser. Mat. 41, No.2, 285-307. Zbl. 362. 57012. English transl.: Math. USSR, Izv. 11,271- 292. Gorbatsevich, V. V. (1979): Splittings of Lie groups and their applications to the study of homogeneous spaces. Izv. Akad. Nauk SSSR, Ser. Mat. 43, No.6, 1127-1157. Zbl. 424. 22006. English transl.: Math. USSR, Izv. 11, 271-292. Gorbatsevich, V. V. (1980): On compact homogeneous spaces of low dimension. In Geom. Metody Zadachakh Algebry Anal. 2, 37-60 (Russian). Zbl. 475. 53043 Gorbatsevich, V. V. (1981a): On compact homogeneous spaces with solvable fundamental group 1. In Geom. Metody Zadachakh Algebry Anal. 1981, 71-87 (Russian) Zbl. 499. 57015 Gorbatsevich, V. V. (1981b): On a fibration of a compact homogeneous space. Tr. Mosk. Mat. O. -va 43,116-141. English transl.: Trans. Mosc. Math. Soc. No.1, 128-157 (1983). Zbl. 525. 57034

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Gorbatsevich, V. V. (1981c): Modifications of transitive actions of Lie groups on compact manifolds and their applications. In Vopr. Teor. Grupp Gomologicheskoj Algebry, 131145 (Russian). Zbl. 506. 57022 Gorbatsevich, V. V. (1981d): Two fibrations of a compact homogeneous space and some applications. Izv. Vyssh. Uchebn. Zaved., Mat. No.6, 73-75 . Zbl. 492. 57013. English transl.: SOy. Math. 25, No.6, 73-75 Gorbatsevich, V. V. (1981e): On compact homogeneous spaces with semi-simple fundamental group I. Sib. Mat. Zh. 22, No.1, 47-67. Zbl. 4R6. 57019. English transl.: Sib. Math. J. 22, 34-49 Gorbatsevich, V. V. (1982): On compact homogeneous spaces with solvable fundamental group II. Vopr. Teor. Grupp Gomologicheskoy Algebry, 13-28 (Russian). Zbl. 583. 57014 Gorbatsevich, V. V. (1983a): On a class of compact homogeneous spaces. bv. Vyssh. Uchebn. Zaved., Mat. 9, 18-21. Zbl. 564. 57019. English transl.: SOy. Math 27, No.9, 18-22 Gorbatsevich, V. V. (1983b): Compact aspherical homogeneous spaces up to a finite covering. Ann. Global. Anal. Geom. 1, No.3, 103-118. Zbl. 535. 57024 Gorbatsevich, V. V. (1985): On compact homogeneous spaces with solvable fundamental group III. Vopr. Teor. Grupp Gomologicheskoj Algebry, 93-103 (Russian) Gorbatsevich, V. V (1986a): On Lie groups with lattices and their properties. Dokl. Akad. Nauk SSSR 287, No 1, 33-37. English transl.: SOy. Math., Dokl. 33,321-325. Zbl. 619. 22014 Gorbatsevich, V. V. (1986b): On compact homogeneous spaces with semi-simple fundamental group II. Sib. Mat. Zh. 37, No.5, 38-49. English transl.: Sib. Math. J. 27, 660-669. Zbl. 644. 57006 Gorbatsevich, V. V (1988): On some classes of homogeneous spaces close to compact. Dokl. Akad. Nauk SSSR 303, No.4, 785-788. English transl.: SOy. Math. Dokl. 38, No.3, 592596. Zbl. 703. 22006 Goto, M., Wang, H.-C. (1978): Non-discrete uniform subgroups of semi-simple Lie groups. Math. Ann. 198, No.4, 259-286. Zbl. 228. 22014 Helgason, S. (1962): Differential Geometry and Symmetric Spaces. Academic Press. New York, London. Zbl. 111, 181 Hermann, R. (1965): Compactification of homogeneous spaces. I. J. Math. Mech. 14, No.4, 655-678. Zbl. 141, 196 Hsiang, W. C., Hsiang, W. Y. (1967): Differentiable actions of compact connected classical groups. I. Am. J. Math. 89, No.3, 705-786. Zbl. 184,272 Hsiang, W. Y. (1975): Cohomology Theory of Topological Transformation Groups. Springer, Berlin. Zbl. 429. 57011 Hsiang, W. Y., Su, J. C. (1968): On the classification of transitive actions on Stiefel manifolds. Trans. Am. Math. Soc. 130, No.2, 322-336. Zbl. 429. 57011 Husemoller, D. (1966): Fibre Bundles. McGraw-Hill, New York. Zbl. 199, 271 Ibragimov, N. Kh. (1983): Transformation Groups in Mathematical Physics. Nauka, Moscow. Zbl. 529. 53014. English transl.: Reidel, Dordrecht 1985 Iwahori, N., Sugiura, M. (1966): A duality theorem for homogeneous manifolds of compact Lie groups. Osaka J. Math. 3, No.1, 139-153. Zbl. 158, 277 Jiinich, K. (1968): Differenzierbare G-Mannigfaltigkeiten. Lect. Notes Math. 59, Springer, Berlin. Zbl. 159, 537 Johnson, R. (1972): Presentation of solvmanifolds. Am. J. Math. 94, No.1, 82-102. Kamerich, B. N. P. (1977): Transitive transformation groups of products of two spheres. Krips Repro. Meppel. Kantor, I. L. (1974): The double ratio of four points and other invariants on homogeneous spaces with parabolic stabilizers. Seminar on Vector and tensor analysis 17, 250-313 (Russian) Karpelevich, F. I. (1953): Surfaces of transitivity of a semi-simple subgroup of the group of motions of a symmetric space. Dokl. Akad. Nauk SSSR 93, 401-404 (Russian)

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Mostow, G. D. (1954): Factor spaces of solvable groups. Ann. Math., II. Ser. 60, No.1, 1-27. Zb!. 57, 261 Mostow, G. D. (1955a): Some new decomposition theorems for semi-simple groups. Mem. Am. Math. Soc. 14, 31-54 . Zb!. 64, 259 Mostow, G. D. (1955b): On covariant fiberings of Klein spaces I. Am. J. Math. 77, No.2, 247-278. Zbl. 67, 160 Mostow, G. D. (1957): Equivariant embed dings in Euclidean space. Ann. Math., II. Ser. 65, No.3, 432-446. Zb!. 80, 167 Mostow, G. D. (1961): On maximal subgroups in real Lie groups. Ann. Math., II. Ser. 74, No.3, 503-517. Zb!. 80, 167 Mostow, G. D. (1962a): Homogeneous spaces with finite invariant measure. Ann. Math., II. Ser. 75, No.1, 17-37. Zbl. 115, 257 Mostow, G. D. (1962b): Covariant fiberings of Klein spaces II. Am. J. Math. 84, No.3, 466-474 . Zb!. 123, 163 Mostow, G. D. (1971): Arithmetic subgroups of groups with radica!' Ann. Math., II. Ser. 93, No.3, 409-438. Zb!. 212, 364 Mostow, G. D. (1975): On the topology of homogeneous spaces of finite measure. Symp. Math. 1st. Naz. Alta Mat. 16, Acad. Press, 375-398. Zb!. 319. 22008 Nagano, T. (1965): Transformation groups on compact symmetric spaces. Trans. Am. Math. Soc. 118,428-453. Zb!. 151, 288 Nakamura, I. (1975): Complex parallelisable manifolds and their small deformations. J. Differ. Geom. 10, No.1, 85-112. Zb!. 297. 32019 Nazaryan, R. O. (1975a): On factorization of simple real Lie groups. Izy. Akad. Nauk Arm. SSR, 10, No.1, 3-22 (Russian). Zb!. 308. 22011 Nazaryan, R. O. (1975b): Minimal factorizations of simple real Lie groups. Izv. Akad. Nauk Arm. SSR, Mat. 10, No.5, 455-477 (Russian). Zbl. 335. 22008 Nazaryan, R. O. (1981): More about factorizations of simple real Lie groups. In Vopr. Teor. Grupp Gomologicheskoj Algebry, 69-79 (Russian) Onishchik, A. L. (1960): Complex envelopes of compact homogeneous spaces. Dok!. Akad. Nauk SSSR 130, No.4, 726-729 (Russian). Zb!. 90, 94 Onishchik, A. L. (1962): Inclusion relations between transitive compact transformation groups. Tr. Mosk. Mat. O-va, 11, 199-242. Zb!. 192, 126. English trans!.: Trans!., II. Ser., Am. Math. Soc. 50, 5-58 (1966). Onishchik, A. L. (1963): On transitive compact transformation groups. Mat. Sb., Nov. Ser. 60, No.4, 447-485. Zbl. 203, 263. English transl.: Transl., II. Ser., Am. Math. Soc. 55, 153-194 (1966). Onishchik, A. L. (1966): On Lie groups, acting transitively on compact manifolds. I, Mat. Sb., Nov. Ser. 71, No.4, 483-494. Zb!. 198, 289. English trans!.: Trans!., II. Ser., Am. Math. Soc. 73, 59-72 (1968) Onishchik, A. L. (1967): On Lie groups, acting transitively on compact manifolds. II, Mat. Sb., Nov. Ser. 74, No.3, 398-416. Zbl. 198, 289. English transl.: Math. USSR, Sb. 3, 373-388 (1968) Onishchik, A. L. (1968): On Lie groups, acting transitively on compact manifolds. III, Mat. Sb., Nov. Ser. 75, No.2, 255-263. Zbl. 198, 290. English trans!.: Math. USSR, Sb. 4, 233-240 (1969) Onishchik, A. L. (1969): Decompositions of reductive Lie groups. Mat. Sb., Nov. Ser. 80, No.4, 553-599. Zbl. 222. 22011. English trans!.: Math. USSR, Sb. 9, 515-554 Onishchik, A. L. (1970): Lie groups which act transitively on Grassmann and Stiefel manifolds. Mat. Sb., Nov. Ser. 83, No.3, 407-428. Zb!. 206, 317. English trans!.: Math. USSR, Sb. 12, 405-427 Onishchik, A. L. (1976): On invariants and almost invariants of compact Lie groups of transformations. Tr. Mosk. Mat. O.-va 35, 235-264. Zbl. 406. 57025. English trans!.: Trans. Mosc. Math. Soc., 35, 237-267

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Sulanke, R., Wintgen, P. (1972): Differentialgeometrie und Faserbiindel. Deutscher Verlag des Wissenschaften, Berlin. Zbl. 327. 53020 Tits, J. (1962): Espaces homogEmes complexes compacts. Comment Math. Helv. 37, No.2, 111-120. Zbl. 108,363 Vinberg, E. B. (1961): The Morozov-Borel theorem for real Lie groups. Dokl. Akad Nauk SSSR 141, No.2, 270-273. English transl.: SOy. Math., Dokl. 2, 1416-1419. Zbl. 112,25 Vinberg, E. B. (1963): Lie groups and homogeneous spaces. Itogi Nauki Tekh., Ser. Algebra, Topologiya. 1962, 5-32 (Russian). Zbl. 132, 22 Vishik, E. Ya. (1973): Lie groups, transitive on simply connected compact manifolds. Mat. Sb., Nov. Ser. 92, No.4, 564--570. Zbl. 289. 22007. English transl.: Math. USSR, Sb. 21, 558-564 Vladimirov, S. A. (1979): Groups of Symmetries of Differential Equations and Relativistic Fields. Atomizdat, Moscow. Zbl. 399. 58021 Wang, H.-C. (1954): Closed manifolds with homogeneous complex structures. Am. J. Math. 76, No.1, 1-32. Zbl.55, 166 Wang, H.-C. (1956): Discrete subgroups of solvable Lie groups. Ann. Math., II. Ser. 64, No.1, 1-19. Zbl. 73, 285 Warner, F. W. (1983): Foundations of Differentiable Manifolds and Lie Groups. Springer, New York. Zbl. 516. 58001 Wasserman, A. (1969): Equivariant differential topology. Topology 8, No.2, 127-150. Zbl. 215, 247 Wells, R. O. (1973): Differential Analysis on Complex Manifolds. Prentice Hall, Englewood Cliffs. Zbl. 262. 32005. 2nd ed. (1980) Springer, New York. Zbl. 435. 32004 Wolf, J. A. (1968): The geometry and structure of isotropy-irreducible homogeneous spaces. Acta Math. 120,59-148. Zbl. 157,521 Wolf, J. A. (1972): Spaces of Constant Curvature. Univ. of California Press, Berkeley. Zbl. 162, 533. 3rd. ed. (1974) Publish or Perish, Boston. Zbl. 281. 53034 Zabotin, Ya. I. (1958a): Semisimple transitive imprimitive groups of the four-dimensional complex space. Izv. Vyssh. Uchebn. Zaved., Mat. 1958, No.4, 67-79 (Russian). Zbl. 125, 17 Zabotin, Ya. I. (1958b): On transitive imprimitive groups with radical in the four-dimensional complex space. Izv. Vyssh. Uchebn. Zaved., Mat. 1958, No.5, 73-85 (Russian). Zbl. 125, 17

Author Index Auslander, L. 169 Betti, E. 167 Birkhoff, G. 61 Bochner, S. 114, 115 Bol, G. 88, 91 Borel, A. 152 Campbell, J. E. 59, 71 Cartan, E. 47, 87, 218 Cayley, A. 89 Coxeter, H. 127 Dufio, M. 64 Dynkin, E. B. 72 Frechet, M. 84, 135 Frobenius, F. 131 Gelfand, 1. M. 64, 65 Gleason, A. 88 Goldie, A. W. 64 Grassmann, H. G. 120 Hausdorff, F. 59, 71 Heisenberg, W. 50 Helgason, S. 46 Hilbert, D. 74, 87, 88, 156 Hirsch, K. A. 201 Hopf, H. 122 Iwasawa, K. 161 Jacobi, C. 32 Karpelevich, F. 1. 136 Killing, W. 183 Kirillov, A. A. 64, 65 Klein, F. 121, 170 Koszul, J.-L. 150 Kreck, M. 186, 193

Levi, E. E. 137 Lie, S. 4, 6, 86, 99, 103, 111, 210, 212, 214 Lobachevski, N. 1. 134 Mobius, A. 164 Maltsev, A. 1. 53, 88, 91, 161 Mann, L. N. 154 Mikhailichenko, G. G. 216 Milovanov, M. B. 170 Montgomery, D. 88, 114, 115, 147, 149, 150, 153, 179 Mostow, G. D. 136, 137, 149, 166, 218 Moufang, Ruth 89, 91 Myers, S. B. 115 Ore, 0. 64 Ostrowski, A. 75 Palais, R. S. 149 Peter, F. 156 Poincare, H. 59, 61 Pontryagin, L. S. 88, 169 Postnikov, N. N. 29, 193 Samelson, H. 137,149,179 . Selberg, A. 190 Serre, J .-P. 200 Steenrod, N. 115 Stein, K. 159 Stiefel, E. 120, 166, 167 Stolz, S. 186, 194 Taylor, B. 68 Tits, J.174, 199 Wang, E. H.-C. 169 Weyl, H. 65, 127, 156 Whitney, H. 166,167 Witt, E. 61 Yang, C.-T. 150 Zippin, L. 88

Subject Index A-structure 114 Absolute value 74 Action of a group 100 asystatic 124 by inner automorphisms 103 by left translations 103 - - by right translations 103 by two sided translations 103 effective 101 free 106 left 100 - - linear 102 primitive 124 right 100 simply transitive 106 systatic 124 - - transitive 103 trivial 100 - of a Lie algebra 211 locally primitive 212 - - locally transitive 212 - - primitive 212 transitive 212 - of a Lie group 12, 100 affine 12 asystatic 124 - - irreducible 140 - - linear 12 linear compact 108 local 104 local globalizable 105 locally effective 101 - - minimal 220 - - primitive 124 proper 149 regular 195 Algebra, Maltsev 91 - universal enveloping 60 - Banach 82 - Bol91 - Heisenberg 50 - Hopf66 - Lie 32 - - free 70

nilpotent 58 semisimple 57 solvable 56 - Moufang-Lie 91 - Weyl65 Bialgebra 66 Bundle 115 - associated 117 - Borel 194 - frame 116 - homogeneous 129 - Hopf 122 - induced 119 - Karpelevich-Mostow 136 - locally trivial 15 - Mostow 168 - natural 190 - of A-structures 218 - of positive densities 117 - principal 116 - structure 197 - tautological 120 - Tits 174 - trivial 15, 116 - universal 120 Canonical coordinates of the first kind 45 - of the second kind 45 Central series decreasing of a Lie group 58 - - algebra 58 Centralizer of a Lie subgroup 40 - an element of a Lie algebra 36 Centre of a Lie algebra 40 - group 40 Commensurability of groups 198 - weak 198 Commutator (ideal) mutual of ideals 55 - (subalgebra) of a Lie algebra 52 - (subgroup) mutual of normal subgroups 55 - (subgroup) of a Lie group 52

Subject Index - higher of a Lie algebra 56 - - of a Lie group 56 - of vector fields 111 Component of a homogeneous manifold - almost simply connected 203 - semisimple 203 - solvable 203 Comultiplication 71 Corank of a manifold 129 Decomposable manifold 219 Deformation of a path 40 Dense winding of the torus 14 Derivation of a Lie algebra interior 50 Derivation of an algebra 35 Diagonal map 66 Differential of a homomorphism of Lie groups 32 - of an action of a Lie group 34 - of the exponential mapping 46 - operator right invariant 67 Dimension, cohomological 200 - Gelfand-Kirillov 65 - virtual 200 Direction vector of a one-parameter subgroup 44 Element almost invariant 102 - invariant 100 - regular of a compact Lie group 153 - - of a tangent algebra 153 - representative 102 - singular of a compact Lie group 153 Enlargement of an action 102 - natural 139 - of type I 145 - of type II 145 - radical 178 Euler characteristic of a group 201 - of a manifold 129 Exponential mapping 45, 77 Extension of structure group 116 J-projectable field 110 Factorization of a group 138 - of a Lie algebra 148 - of a Lie group 141 global 142 irreducible 143 maximal 178 trivial 143 Fibre bundle 115 - of a bundle 115 Fixed point 100

Flag manifold 148 Flow 102 - local 108 Formal group law 78 Function representative 107 Functor Lie 29 79 G-bundle 118 - homogeneous 129 - trivial 119 G-space 101 - analytic 101 - differentiable 101 - equiorbital 151 - topological 101 Gauge transformation 82 Geodesic loop 89 Globalization of a local action 105 Group formal 78 - isotropy 105 - - linear 106 - Lie 6 abelian 43 abelian complex 43 abelian real 43 aspherical 204 Banach (Hilbert) 81 complex 6 linear 8 - - local 11 nilpotent 58 of transformations 101 of type (I) 171 p-adic 75 semi-simple 204 solvable 56 standard 76 universal covering 26 vector 7 - Lie-Frechet 84 - Lie-Frechet tame 85 - loop 81 - model 121 - of components of a Lie group 22 - of covering transformations 26 - of flows 81 - of interior automorphisms of a Lie algebra 49 - - of a Lie group 49 - of transformations 101 - semi-simple 214 - topological 86 - virtually torsion free 200 - Wang 169

233

234

Subject Index

Homogeneous manifold 123 Homomorphism of formal groups 78 - of Lie groups 9 - - covering 24 Homotopical characteristic 129 ILB-group (strong) 86 ILH-group (strong) 86 Inclusion of actions 102 - improper 102 - proper 102 Isomorphic actions 101 - local Lie groups 11 Isomorphism of group actions 101 - of homogeneous bundles 130 - of Lie Groups 9 Iwasawa Manifold 161 Jacobi identity 32 Kernel non-effectiveness 101 - of action 101 Klein model 121 Largest nilpotent ideal 58 - normal Lie subgroup 59 Lie bracket 111 - derivative 109 Lifting of an action 155 Linear isotropy representation 106 - representation of a Lie group 9 Linearlization of an action 103 Localization of an action 104 Locally isomorphic Lie algebra actions 211 Lie groups 11 - local actions of Lie groups 211 - triples of Lie groups 141 Locally similar actions of Lie algebras 211 - local actions of Lie groups 211 - subalgebras of the Lie algebra of vector fields 211 Loop 88 - alternative 89 - analytic 89 Bol88 - disassociative 89 - local 89 - monoassociative 89 - Moufang 89 Maltsev closure 53 Manifold homogeneously decomposable 219 Mapping equivariant 101

Morphism of actions 101 - of homogeneous bundles 130 - of local actions 104 Multiplication 66 Nilmanifold 160 Norm 74 - non-archimedean 74 - ultrametric 74 Normalizer of a subspace in an algebra 37 Normed field 74 - non-archimedean field 74 Number Coxeter 127 - of ends of a group 202 One-paremeter subgroup 44 Orbit 105 - exceptional 153 - principal 153 singular 153 Part of a fundamental group semi-simple 200 - solvable 200 Path 37 Poincare Polynomial 126 - series 126 Primitive element of a bialgebra 66 Product direct of Lie groups 7 - fibred 117 - semi-direct 20 Projection map of a fibre bundle 115 Pseudo orthogonal matrix 23 Pseudo unitary matrix 23 Quotient group Lie 17 - map 15 Radical of a group 199 - of a Lie algebra 57 - of a Lie group 57 Rank of a compact Lie group 127 - of a manifold 129 Reduction of action 102 - of structure group 116 Representation adjoint of a Lie algebra 36 - - of a Lie group 35 - induced 131 Restriction of a local Lie group 11 - of an action 104 Section of a bundle 117 Similar group actions 102 Similitude of group actions 102

Subject Index - of local actions of Lie groups 104 Skew-field enveloping 64 - Lie 64 Slice 150 Solvmanifold 160 - complex 171 Space Frt!chet 84 - - tame 85 - homogeneous 103 elementary 208 reductive 132 semi-simple 204 solvable 204 - Lobachevski 134 Stabilizer of a point 105 Subalgebra elliptic 114 - of finite order 114 - primitive 124 - reductive 160 - stabilizer 212 - triangular 173 - uniform 175 Subgroup Ad-algebraic 172 - arithmetical 162 - cocompact 172 - - in a Lie group 177 - Lie 7 - - virtual 38 - of maximal exponent 127 - primitive 124 - stabilizer (isotropy) 105 - triangular 173 - uniform 172

235

Sum semi-direct 51 - of Lie algebras 51 System dynamical with continuous time 102 - of generators free 70 t-subalgebra 173 t-subgroup 173 Tangent algebra of a formal group 79 - of a Lie group 30, 76, 86 - of a local analytic loop 90 Theorem Cartan's 47 - Mostow's Structure 167 Topological G-module 135 Torus 7 Total space of a fibre bundle 115 Tower of principal fibrations 164 Transformation inifinitesimal 109 Type orbit 106 - principal 153 Unification 138 Unit of a loop 88 Universal covering 26 Valuation ring 75 Vector field, fundamental 112 - right-invariant 31, 113 - /-projectable 110 - complete 109 - invariant 110 Velocity field of an action 34 - of a path in a Lie group 37

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