Introduction | p. xi |
Terminology, conventions, and notation | p. xix |
Constructions, examples, and structure theory | p. 1 |
Overview of pseudo-reductivity | p. 3 |
Comparison with the reductive case | p. 3 |
Elementary properties of pseudo-reductive groups | p. 11 |
Preparations for the standard construction | p. 16 |
The standard construction and examples | p. 25 |
Main result | p. 34 |
Weil restriction and fields of definition | p. 36 |
Root groups and root systems | p. 43 |
Limits associated to 1 -parameter subgroups | p. 43 |
Pseudo-parabolic subgroups | p. 59 |
Root groups in pseudo-reductive groups | p. 67 |
Representability of automorphism functors | p. 78 |
Basic structure theory | p. 86 |
Perfect normal subgroups of pseudo-reductive groups | p. 86 |
Root datum for pseudo-reductive groups | p. 94 |
Unipotent groups associated to semigroups of roots | p. 99 |
Bruhat decomposition and Levi subgroups | p. 114 |
Classification of pseudo-parabolic subgroups | p. 130 |
Standard presentations and their applications | p. 147 |
Variation of (G′, k′/k, T′, C) | p. 149 |
Absolutely simple and simply connected fibers | p. 149 |
Uniqueness of (G′, k′/k) | p. 154 |
Ubiquity of the standard construction | p. 162 |
Main theorem and central extensions | p. 162 |
Properties of standardness and standard presentations | p. 170 |
A standardness criterion | p. 179 |
Classification results | p. 191 |
The A1-case away from characteristic 2 | p. 192 |
Types A2 and G2 away from characteristic 3 | p. 198 |
General cases away from characteristics 2 and 3 | p. 203 |
General classification and applications | p. 215 |
The exotic constructions | p. 217 |
Calculations in characteristics 2 and 3 | p. 217 |
Basic exotic pseudo-reductive groups | p. 228 |
Algebraic and arithmetic aspects of basic exotic pseudo-reductive groups | p. 240 |
Preparations for classification in characteristics 2 and 3 | p. 255 |
Further properties of basic exotic pseudo-reductive groups | p. 255 |
Exceptional and exotic pseudo-reductive groups | p. 260 |
The absolutely pseudo-simple groups in characteristic 2 | p. 279 |
TypeA1 | p. 280 |
Root groups and birational group laws | p. 290 |
Construction of absolutely pseudo-simple groups with a non-reduced root system | p. 299 |
Classification of absolutely pseudo-simple groups with a non-reduced root system | p. 318 |
General case | p. 342 |
Factors with non-reduced root system and the generalized standard construction | p. 342 |
Classification via generalized standard groups | p. 351 |
Applications | p. 358 |
Maximal tori in pseudo-reductive groups | p. 358 |
Pseudo-semisimplicity | p. 364 |
Unirationality | p. 368 |
Structure of root groups and pseudo-parabolic subgroups | p. 372 |
Appendices | p. 389 |
Background in linear algebraic groups | p. 391 |
Review of definitions | p. 392 |
Some results from the general theory | p. 398 |
Frobenius morphisms and non-affine groups | p. 402 |
Split reductive groups: Existence, Isomorphism, and Isogeny Theorems | p. 407 |
Weil restriction generalities | p. 422 |
Groups without Levi subgroups | p. 441 |
Lie algebras and Weil restriction | p. 446 |
Lie algebras and groups of multiplicative type | p. 457 |
Tits' work on unipotent groups in nonzero characteristic | p. 473 |
Subgroups of vector groups | p. 473 |
Wound unipotent groups | p. 479 |
The cckp-kernel | p. 482 |
Torus actions on unipotent groups | p. 485 |
Rational conjugacy in connected groups | p. 494 |
Pseudo-completeness | p. 494 |
Conjugacy results in the smooth affine case | p. 504 |
Split unipotent subgroups of pseudo-reductive groups | p. 511 |
Beyond the smooth affine case | p. 519 |
References | p. 525 |
Index | p. 527 |
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