Actions and Invariants of Algebraic Groups

Actions and Invariants of Algebraic Groups......Page 4
Contents......Page 6
Preface......Page 11
Acknowledgments......Page 13
Enumeration of items and cross references......Page 15
APPENDIX: Basic definitions and results......Page 434
Bibliography......Page 444
Glossary of Notations......Page 451
1. Introduction......Page 17
2.1. Ring and field extensions......Page 19
2.2. Hilbert’s Nullstellensatz......Page 23
2.3. Separability......Page 25
2.4. Faithfully flat ring extensions......Page 28
2.5. Regular local rings......Page 29
3.1. Basic definitions......Page 30
3.2. The Zariski topology......Page 33
3.3. Polynomial maps. Morphisms......Page 35
4.1. Sheaves on topological spaces......Page 39
4.2. The maximal spectrum......Page 42
4.3. Affine algebraic varieties......Page 43
4.4. Algebraic varieties......Page 50
4.5. Morphisms of algebraic varieties......Page 61
4.6. Complete varieties......Page 67
4.7. Singular points and normal varieties......Page 69
5. Deeper results on morphisms......Page 72
6. Exercises......Page 80
1. Introduction......Page 88
2. Definitions and basic concepts......Page 89
3. The theorems of F. Engel and S. Lie......Page 93
4. Semisimple Lie algebras......Page 99
5. Cohomology of Lie algebras......Page 104
6. The theorems of H. Weyl and F. Levi......Page 112
7. p–Lie algebras......Page 114
8. Exercises......Page 116
1. Introduction......Page 121
2. Definitions and basic concepts......Page 122
3. Subgroups and homomorphisms......Page 128
4. Actions of affine groups on algebraic varieties......Page 130
5. Subgroups and semidirect products......Page 135
6. Exercises......Page 140
1. Introduction......Page 145
2. Hopf algebras and algebraic groups......Page 146
3. Rational G–modules......Page 153
4. Representations of SL2......Page 163
5. Characters and semi–invariants......Page 165
6. The Lie algebra associated to an affine algebraic group......Page 168
7. Explicit computations......Page 171
8. Exercises......Page 179
1. Introduction......Page 186
2. The Jordan decomposition of a single operator......Page 187
3. The Jordan decomposition of an algebra homomorphism and of a derivation......Page 192
4. Jordan decomposition for coalgebras......Page 194
5. Jordan decomposition for an affine algebraic group......Page 200
6. Unipotency and semisimplicity......Page 204
7. The solvable and the unipotent radical......Page 211
8. Structure of solvable groups......Page 217
9.1. The general linear group GLn......Page 223
9.2. The special linear group SLn (case A)......Page 224
9.3. The projective general linear group PGLn(K) (case A)......Page 225
9.4. The special orthogonal group SOn (cases B,D)......Page 226
9.5. The symplectic group Spn, n = 2m (case C)......Page 227
10. Exercises......Page 228
1. Introduction......Page 232
2. Actions: examples and first properties......Page 233
3. Basic facts about the geometry of the orbits......Page 237
4. Categorical and geometric quotients......Page 240
5. The subalgebra of invariants......Page 251
6. Induction and restriction of representations......Page 255
7. Exercises......Page 261
1. Introduction......Page 266
2. Embedding H–modules inside G–modules......Page 267
3. Definition of subgroups in terms of semi–invariants......Page 271
4. The coset space G/H as a geometric quotient......Page 279
5. Quotients by normal subgroups......Page 281
6. Applications and examples......Page 284
7. Exercises......Page 289
1. Introduction......Page 292
2. Correspondence between subgroups and subalgebras......Page 294
3. Algebraic Lie algebras......Page 300
4. Exercises......Page 304
1. Introduction......Page 307
2. Linear and geometric reductivity......Page 309
3. Examples of linearly and geometrically reductive groups......Page 320
4. Reductivity and the structure of the group......Page 326
5. Reductive groups are linearly reductive in characteristic zero......Page 330
6. Exercises......Page 332
1. Introduction......Page 335
2. Basic definitions......Page 336
3. Induction and observability......Page 340
4. Split and strong observability......Page 342
5. The geometric characterization of observability......Page 351
6. Exercises......Page 353
1. Introduction......Page 356
2. Geometric reductivity and observability......Page 357
3. Exact subgroups......Page 358
4. From quasi–affine to affine homogeneous spaces......Page 359
5. Exactness, Reynolds operators, total integrals......Page 361
6. Affine homogeneous spaces and exactness......Page 364
7. Affine homogeneous spaces and reductivity......Page 367
8. Exactness and integrals for unipotent groups......Page 368
9. Exercises......Page 371
1. Introduction......Page 374
2. A counterexample to Hilbert’s 14th problem......Page 377
3. Reductive groups and finite generation of invariants......Page 386
4. V. Popov’s converse to Nagata’s theorem......Page 390
5. Partial positive answers to Hilbert’s 14th problem......Page 392
6. Geometric characterization of Grosshans pairs......Page 398
7. Exercises......Page 400
1. Introduction......Page 404
2. Actions by reductive groups: the categorical quotient......Page 405
3. Actions by reductive groups: the geometric quotient......Page 411
4. Canonical forms of matrices: a geometric perspective......Page 415
5. Rosenlicht’s theorem......Page 418
6. Further results on invariants of finite groups......Page 421
6.1. Invariants of graded algebras......Page 422
6.2. Polynomial subalgebras of polynomial algebras......Page 424
6.3. The case of a group generated by reflections......Page 428
6.4. The degree of the fundamental invariants for a finite group......Page 430
7. Exercises......Page 431
2.3. Linear algebra......Page 435
2.4. Group theory......Page 436
3. Rings and modules......Page 438
4. Representations......Page 442