Algebraic transformation groups - an introduction

271 Pages·2018·3.398 MB·English

by  Hanspeter Kraft

ALGEBRAIC
TRANSFORMATION GROUPS
- AN INTRODUCTION -
Hanspeter Kraft
(Preliminary version from February 2009, updated in June 2011, July 2014,
October 2016, January 2017)
Universita¨t Basel
Departement Mathematik und Informatik
Spiegelgasse 1, CH-4051 Basel, Switzerland
Email address: [email protected]
URL: http://www.math.unibas.ch/kraft
2010 Mathematics Subject Classification. Primary
Key words and phrases. algebraic groups, group actions, transformation groups,
representation theory, invariants, algebraic quotient
To Renate, Marcel, Christian and Claudine

Contents
Preface xiii
Chapter I. First Examples and Basic Concepts 1
Introduction 1
1. Elementary Euclidean Geometry 2
1.1. Triangles 2
1.2. Invariants 3
1.3. Congruence classes 4
1.4. Orbit space and quotient map 4
1.5. Summary 6
2. Symmetric Product and Symmetric Functions 7
2.1. Symmetric product 7
2.2. Symmetric functions 7
2.3. Roots of polynomials 9
3. Quadratic Forms 10
3.1. Equivalence classes 10
3.2. Closures of equivalence classes 11
3.3. Other equivalence relations 12
4. Conjugacy Classes of Matrices 13
4.1. Adjoint representation 13
4.2. The geometry of π: M →Cn 14
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4.3. Cyclic matrices 15
4.4. The nilpotent cone 15
5. Invariants of Several Vectors 16
5.1. Pairs of vectors 16
5.2. The null fiber 17
5.3. Vector bundles over P1 18
5.4. Invariants of several vectors 19
6. Nullforms 20
6.1. Binary forms 20
6.2. The null cone of R 22
5
6.3. A geometric picture of N 23
5
7. Deformations and Associated Cone 25
7.1. The associated cone 25
7.2. Conjugacy classes of matrices 26
7.3. The case of binary forms of degree five 27
8. Ternary Cubics 29
8.1. Normal forms 29
8.2. Classification with respect to SL 31
3
8.3. Nullforms and degenerations 32
8.4. Invariants under SL 33
3
8.5. Some computations 34
Exercises 35
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viii CHAPTER.CONTENTS
Chapter II. Algebraic Groups 37
Introduction 37
1. Basic Definitions 38
1.1. Linear algebraic groups 38
1.2. Isomorphisms and products 40
1.3. Comultiplication and coinverse 41
1.4. Connected component 42
1.5. Exercises 43
2. Homomorphisms and Exponential Map 45
2.1. Homomorphisms 45
2.2. Characters and the character group 47
2.3. Normalizer, centralizer, and center 48
2.4. Commutator subgroup 49
2.5. Exponential map 50
2.6. Unipotent elements 51
2.7. Exercises 52
3. The Classical Groups 54
3.1. General and special linear groups 54
3.2. Orthogonal groups 56
3.3. Symplectic groups 58
3.4. Exercises 60
4. The Lie Algebra of an Algebraic Group 60
4.1. Lie algebras 60
4.2. The Lie algebra of GL 61
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4.3. The classical Lie algebras 62
4.4. The adjoint representation 63
4.5. Invariant vector fields 65
Exercises 66
Chapter III. Group Actions and Representations 67
Introduction 67
1. Group Actions on Varieties 68
1.1. G-Varieties 68
1.2. Fixed Points, Orbits and Stabilizers 68
1.3. Orbit map and dimension formula 70
1.4. Exercises 71
2. Linear Actions and Representations 71
2.1. Linear representation 71
2.2. Construction of representations and G-homomorphisms 73
2.3. The regular representation 74
2.4. Subrepresentations of the regular representation 76
2.5. Exercises 78
3. Tori and Diagonalizable Groups 78
3.1. C∗-actions and quotients 78
3.2. Tori 81
3.3. Diagonalizable groups 82
3.4. Characterization of tori and diagonalizable groups 82
3.5. Classification of diagonalizable groups 84
3.6. Invariant rational functions 85
3.7. Exercises 87
4. Jordan Decomposition and Commutative Algebraic Groups 88
4.1. Jordan decomposition 88
4.2. Semisimple elements 88
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4.3. Commutative algebraic groups 89
4.4. Exercises 90
5. The Correspondence between Groups and Lie Algebras 91
5.1. The differential of the orbit map 91
5.2. Subgroups and subalgebras 92
5.3. Representations of Lie algebras 93
5.4. Vector fields on G-varieties 94
5.5. G-action on vector fields 97
5.6. Jordan decomposition in the Lie algebra 98
5.7. Invertible functions and characters 99
5.8. C+-actions and locally nilpotent vector fields 100
Exercises 103
Chapter IV. Invariants and Algebraic Quotients 105
Introduction 106
1. Isotypic Decomposition 108
1.1. Completely reducible representations 108
1.2. Endomorphisms of semisimple modules 109
1.3. Isotypic decomposition 110
2. Invariants and Algebraic Quotients 112
2.1. Linearly reductive groups 112
2.2. The coordinate ring of a linearly reductive group 113
2.3. Hilbert’s Finiteness Theorem 114
2.4. Algebraic quotient 115
2.5. Properties of quotients 116
2.6. Some consequences 117
2.7. The case of finite groups 118
3. The Quotient Criterion and Applications 120
3.1. Properties of quotients 120
3.2. Some examples revisited 121
3.3. Cosets and quotient groups 123
3.4. A criterion for quotients 123
4. The First Fundamental Theorem for GL 125
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4.1. A Classical Problem 125
4.2. First Fundamental Theorem 126
4.3. A special case 127
4.4. Orbits in L(U,V) 127
4.5. Degenerations of orbits 129
4.6. The subgroup H 131
ρ
4.7. Structure of the fiber F 132
ρ
5. Sheets, General Fiber and Null Fiber 134
5.1. Sheets 135
5.2. Finitely many orbits 137
5.3. The associated cone 138
5.4. The coordinate ring of the associated cone 140
5.5. Reducedness and normality 142
6. The Variety of Representations of an Algebra 143
6.1. The variety Modn 143
A
6.2. Geometric properties 145
6.3. Degenerations 146
6.4. Tangent spaces and extensions 147
7. Structure of the Quotient 149
7.1. Inheritance properties 149
x CHAPTER.CONTENTS
7.2. Singularities in the quotient 149
7.3. Smooth quotients 150
7.4. Semi-continuity statements 151
7.5. Generic fiber 152
7.6. A finiteness theorem 153
8. Quotients for Non-Reductive Groups 154
Exercises 154
Chapter V. Representation Theory and U-Invariants 155
1. Representations of Linearly Reductive Groups 155
1.1. Commutative and Diagonalizable Groups 155
1.2. Unipotent Groups 156
1.3. Solvable Groups 156
1.4. Representation theory of GL 156
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1.5. Representation theory of reductive groups 156
2. Characterization of Reductive Groups 156
2.1. Definitions 156
2.2. Images and kernels 157
2.3. Semisimple groups 158
2.4. The classical groups 159
2.5. Reductivity of the classical groups 160
3. Hilbert’s Criterion 160
3.1. One-parameter subgroups 160
3.2. Torus actions 160
3.3. Hilbert’s Criterion for GL 160
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3.4. Hilbert’s Criterion for reductive groups 160
4. U-Invariants and Normality Problems 160
Exercises 160
Appendix A. Basics from Algebraic Geometry 161
1. Affine Varieties 163
1.1. Regular functions 163
1.2. Zero sets and Zariski topology 164
1.3. Hilbert’s Nullstellensatz 166
1.4. Affine varieties 169
1.5. Special open sets 171
1.6. Decomposition into irreducible components 172
1.7. Rational functions and local rings 174
2. Morphisms 176
2.1. Morphisms and comorphisms 176
2.2. Images, preimages and fibers 178
2.3. Dominant morphisms and degree 180
2.4. Rational varieties and Lu¨roth’s Theorem 181
2.5. Products 182
2.6. Fiber products 183
3. Dimension 184
3.1. Definitions and basic results 184
3.2. Finite morphisms 186
3.3. Krull’s principal ideal theorem 190
3.4. Decomposition Theorem and dimension formula 192
3.5. Constructible sets 194
3.6. Degree of a morphism 195
3.7. Mo¨bius transformations 196
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4. Tangent Spaces, Differentials, and Vector Fields 196
4.1. Zariski tangent space 196
4.2. Tangent spaces of subvarieties 198
4.3. R-valued points and epsilonization 199
4.4. Nonsingular varieties 200
4.5. Tangent bundle and vector fields 201
4.6. Differential of a morphism 204
4.7. Epsilonization 206
4.8. Tangent spaces of fibers 206
4.9. Morphisms of maximal rank 207
4.10. Associated graded algebras 210
4.11. m-adic completion 212
5. Normal Varieties and Divisors 213
5.1. Normality 213
5.2. Integral closure and normalization 214
5.3. Analytic normality 217
5.4. Discrete valuation rings and smoothness 217
5.5. The case of curves 219
5.6. Zariski’s Main Theorem 220
5.7. Complete intersections 223
5.8. Divisors 223
Exercises 225
Appendix B. The Strong Topology on Complex Affine Varieties 233
1. C-Topology on Varieties 234
1.1. Smooth points 234
1.2. Proper morphisms 234
1.3. Connectedness 235
1.4. Holomorphic functions satisfying an algebraic equation 235
1.5. Closures in Zariski- and C-topology 236
2. Reductivity of the Classical Groups 236
2.1. Maximal compact subgroups 236
Exercises 236
Appendix C. Fiber Bundles, Slice Theorem and Applications 237
1. Introduction: Local Cross Sections and Slices 238
1.1. Free actions and cross sections 238
1.2. Associated bundles and slices 238
2. Flat and E´tale Morphisms 239
2.1. Unramified and ´etale morphisms 240
2.2. Standard ´etale morphisms 241
2.3. E´tale base change 244
3. Fiber Bundles and Principal Bundles 246
3.1. Additional structures, s-varieties 247
3.2. Fiber bundles 247
3.3. Principal bundles 249
Bibliography 251
Index 255