Actions and Invariants of Algebraic Groups

453 Pages·2005·2.064 MB·English

by  Walter Ferrer Santos, Alvaro Rittatore

Actions and Invariants
of Algebraic Groups
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Actions and Invariants
of Algebraic Groups
Walter Ferrer Santos
Universidad de la República
Montevideo, Uruguay
Alvaro Rittatore
Universidad de la República
Montevideo, Uruguay
Boca Raton London New York Singapore
Published in 2005 by
CRC Press
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Contents
Preface xi
Enumeration of items and cross references xv
Chapter 1. Algebraic Geometry 1
1. Introduction 1
2. Commutative algebra 3
2.1. Ring and field extensions 3
2.2. Hilbert’s Nullstellensatz 7
2.3. Separability 9
2.4. Faithfully flat ring extensions 12
2.5. Regular local rings 13
3. Algebraic subsets of the affine space 14
3.1. Basic definitions 14
3.2. The Zariski topology 17
3.3. Polynomial maps. Morphisms 19
4. Algebraic varieties 23
4.1. Sheaves on topological spaces 23
4.2. The maximal spectrum 26
4.3. Affine algebraic varieties 27
4.4. Algebraic varieties 34
4.5. Morphisms of algebraic varieties 45
4.6. Complete varieties 51
4.7. Singular points and normal varieties 53
5. Deeper results on morphisms 56
6. Exercises 64
Chapter 2. Lie algebras 73
1. Introduction 73
2. Definitions and basic concepts 74
3. The theorems of F. Engel and S. Lie 78
4. Semisimple Lie algebras 84
v
vi
5. Cohomology of Lie algebras 89
6. The theorems of H. Weyl and F. Levi 97
7. p–Lie algebras 99
8. Exercises 101
Chapter 3. Algebraic groups: basic definitions 107
1. Introduction 107
2. Definitions and basic concepts 108
3. Subgroups and homomorphisms 114
4. Actions of affine groups on algebraic varieties 116
5. Subgroups and semidirect products 121
6. Exercises 126
Chapter 4. Algebraic groups: Lie algebras and representations 131
1. Introduction 131
2. Hopf algebras and algebraic groups 132
3. Rational G–modules 139
4. Representations of SL 149
2
5. Characters and semi–invariants 151
6. The Lie algebra associated to an affine algebraic group 154
7. Explicit computations 157
8. Exercises 165
Chapter 5. Algebraic groups: Jordan decomposition and applications173
1. Introduction 173
2. The Jordan decomposition of a single operator 174
3. The Jordan decomposition of an algebra homomorphism and of
a derivation 179
4. Jordan decomposition for coalgebras 181
5. Jordan decomposition for an affine algebraic group 187
6. Unipotency and semisimplicity 191
7. The solvable and the unipotent radical 198
8. Structure of solvable groups 204
9. The classical groups 210
9.1. The general linear group GL 210
n
vii
9.2. The special linear group SL (case A) 211
n
9.3. The projective general linear group PGL (k) (case A) 212
n
9.4. The special orthogonal group SO (cases B,D) 213
n
9.5. The symplectic group Sp , n=2m (case C) 214
n
10. Exercises 215
Chapter 6. Actions of algebraic groups 219
1. Introduction 219
2. Actions: examples and first properties 220
3. Basic facts about the geometry of the orbits 224
4. Categorical and geometric quotients 227
5. The subalgebra of invariants 238
6. Induction and restriction of representations 242
7. Exercises 248
Chapter 7. Homogeneous spaces 253
1. Introduction 253
2. Embedding H–modules inside G–modules 254
3. Definition of subgroups in terms of semi–invariants 258
4. The coset space G/H as a geometric quotient 266
5. Quotients by normal subgroups 268
6. Applications and examples 271
7. Exercises 276
Chapter 8. Algebraic groups and Lie algebras in characteristic zero 279
1. Introduction 279
2. Correspondence between subgroups and subalgebras 281
3. Algebraic Lie algebras 287
4. Exercises 291
Chapter 9. Reductivity 295
1. Introduction 295
2. Linear and geometric reductivity 297
3. Examples of linearly and geometrically reductive groups 308
4. Reductivity and the structure of the group 314
viii
5. Reductive groups are linearly reductive in characteristic zero 318
6. Exercises 320
Chapter 10. Observable subgroups of affine algebraic groups 323
1. Introduction 323
2. Basic definitions 324
3. Induction and observability 328
4. Split and strong observability 330
5. The geometric characterization of observability 339
6. Exercises 341
Chapter 11. Affine homogeneous spaces 345
1. Introduction 345
2. Geometric reductivity and observability 346
3. Exact subgroups 347
4. From quasi–affine to affine homogeneous spaces 348
5. Exactness, Reynolds operators, total integrals 350
6. Affine homogeneous spaces and exactness 353
7. Affine homogeneous spaces and reductivity 356
8. Exactness and integrals for unipotent groups 357
9. Exercises 360
Chapter 12. Hilbert’s 14th problem 363
1. Introduction 363
2. A counterexample to Hilbert’s 14th problem 366
3. Reductive groups and finite generation of invariants 375
4. V. Popov’s converse to Nagata’s theorem 379
5. Partial positive answers to Hilbert’s 14th problem 381
6. Geometric characterization of Grosshans pairs 387
7. Exercises 389
Chapter 13. Quotients 393
1. Introduction 393
2. Actions by reductive groups: the categorical quotient 394
3. Actions by reductive groups: the geometric quotient 400
ix
4. Canonical forms of matrices: a geometric perspective 404
5. Rosenlicht’s theorem 407
6. Further results on invariants of finite groups 410
6.1. Invariants of graded algebras 411
6.2. Polynomial subalgebras of polynomial algebras 413
6.3. The case of a group generated by reflections 417
6.4. The degree of the fundamental invariants for a finite group 419
7. Exercises 420
APPENDIX. Basic definitions and results 423
1. Introduction 423
2. Notations 423
2.1. Category theory 423
2.2. General topology 424
2.3. Linear algebra 424
2.4. Group theory 425
3. Rings and modules 427
4. Representations 431
Bibliography 433
Glossary of Notations 441