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 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 SL2 ............................................... 149 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 applications ........ 173 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 GLn .................................... 210 9.2. The special linear group SLn (case A) ........................... 211 9.3. The projective general linear group PGLn (k) (case A) ........... 212 9.4. The special orthogonal group SOn (cases B,D) .................... 213 9.5. The symplectic group Spn, n = 2m (case C) ....................... 214 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 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-afnne 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 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 Author Index .............................................................. 445 Index ..................................................................... 447