Pseudo-reductive groups

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Why go beyond reductive groups?......Page 13
Imperfect base fields......Page 14
Structure theory......Page 15
Main results and applications......Page 16
Plan of the monograph......Page 19
Acknowledgments......Page 20
Terminology, conventions, and notation......Page 21
Part I: Constructions, examples, and structure theory......Page 23
1.1 Comparison with the reductive case......Page 25
1.2 Elementary properties of pseudo-reductive groups......Page 33
1.3 Preparations for the standard construction......Page 38
1.4 The standard construction and examples......Page 47
1.5 Main result......Page 56
1.6 Weil restriction and fields of definition......Page 58
2.1 Limits associated to 1-parameter subgroups......Page 65
2.2 Pseudo-parabolic subgroups......Page 81
2.3 Root groups in pseudo-reductive groups......Page 89
2.4 Representability of automorphism functors......Page 100
3.1 Perfect normal subgroups of pseudo-reductive groups......Page 108
3.2 Root datum for pseudo-reductive groups......Page 116
3.3 Unipotent groups associated to semigroups of roots......Page 121
3.4 Bruhat decomposition and Levi subgroups......Page 137
3.5 Classification of pseudo-parabolic subgroups......Page 152
Part II: Standard presentations and their applications......Page 169
4.1 Absolutely simple and simply connected fibers......Page 171
4.2 Uniqueness of (G\', k\'/k)......Page 176
5.1 Main theorem and central extensions......Page 184
5.2 Properties of standardness and standard presentations......Page 192
5.3 A standardness criterion......Page 201
6 Classification results......Page 213
6.1 The A1-case away from characteristic 2......Page 214
6.2 Types A2 and G2 away from characteristic 3......Page 220
6.3 General cases away from characteristics 2 and 3......Page 225
Part III: General classification and applications......Page 237
7.1 Calculations in characteristics 2 and 3......Page 239
7.2 Basic exotic pseudo-reductive groups......Page 250
7.3 Algebraic and arithmetic aspects of basic exotic pseudo-reductive groups......Page 262
8.1 Further properties of basic exotic pseudo-reductive groups......Page 277
8.2 Exceptional and exotic pseudo-reductive groups......Page 282
9 The absolutely pseudo-simple groups in characteristic 2......Page 301
9.1 Type A1......Page 302
9.2 Root groups and birational group laws......Page 312
9.3 Construction of absolutely pseudo-simple groups with a non-reduced root system......Page 321
9.4 Classification of absolutely pseudo-simple groups with a non-reduced root system......Page 340
10.1 Factors with non-reduced root system and the generalized standard construction......Page 364
10.2 Classification via generalized standard groups......Page 373
11.1 Maximal tori in pseudo-reductive groups......Page 380
11.2 Pseudo-semisimplicity......Page 386
11.3 Unirationality......Page 390
11.4 Structure of root groups and pseudo-parabolic subgroups......Page 394
Part IV: Appendices......Page 411
Appendix A: Background in linear algebraic groups......Page 413
A.1 Review of definitions......Page 414
A.2 Some results from the general theory......Page 420
A.3 Frobenius morphisms and non-affine groups......Page 424
A.4 Split reductive groups: Existence, Isomorphism, and Isogeny Theorems......Page 429
A.5 Weil restriction generalities......Page 444
A.6 Groups without Levi subgroups......Page 463
A.7 Lie algebras and Weil restriction......Page 468
A.8 Lie algebras and groups of multiplicative type......Page 479
B.1 Subgroups of vector groups......Page 495
B.2 Wound unipotent groups......Page 501
B.3 The cckp-kernel......Page 504
B.4 Torus actions on unipotent groups......Page 507
C.1 Pseudo-completeness......Page 516
C.2 Conjugacy results in the smooth affine case......Page 526
C.3 Split unipotent subgroups of pseudo-reductive groups......Page 533
C.4 Beyond the smooth affine case......Page 541
References......Page 547
Index......Page 549