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Linear Algebraic Groups and Finite Groups of Lie Type.
The first textbook on the subgroup structure, in particular maximal subgroups, for both algebraic and finite groups of Lie type.
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2011.
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| Colección: | Cambridge Studies in Advanced Mathematics, 133.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title; Copyright; Contents; Preface; Tables; Notation; PART I LINEAR ALGEBRAIC GROUPS; 1 Basic concepts; 1.1 Linear algebraic groups and morphisms; 1.2 Examples of algebraic groups; 1.3 Connectedness; 1.4 Dimension; 2 Jordan decomposition; 2.1 Decomposition of endomorphisms; 2.2 Unipotent groups; 3 Commutative linear algebraic groups; 3.1 Jordan decomposition of commutative groups; 3.2 Tori, characters and cocharacters; 4 Connected solvable groups; 4.1 The Lie-Kolchin theorem; 4.2 Structure of connected solvable groups; 5 G-spaces and quotients; 5.1 Actions of algebraic groups.
- 5.2 Existence of rational representations6 Borel subgroups; 6.1 The Borel fixed point theorem; 6.2 Properties of Borel subgroups; 7 The Lie algebra of a linear algebraic group; 7.1 Derivations and differentials; 7.2 The adjoint representation; 8 Structure of reductive groups; 8.1 Root space decomposition; 8.2 Semisimple groups of rank 1; 8.3 Structure of connected reductive groups; 8.4 Structure of semisimple groups; 9 The classification of semisimple algebraic groups; 9.1 Root systems; 9.2 The classification theorem of Chevalley; 10 Exercises for Part I.
- PART II SUBGROUP STRUCTURE AND REPRESENTATION THEORY OF SEMISIMPLE ALGEBRAIC GROUPS11 BN-pairs and Bruhat decomposition; 11.1 On the structure of B; 11.2 Bruhat decomposition; 12 Structure of parabolic subgroups, I; 12.1 Parabolic subgroups; 12.2 Levi decomposition; 13 Subgroups of maximal rank; 13.1 Subsystem subgroups; 13.2 The algorithm of Borel and de Siebenthal; 14 Centralizers and conjugacy classes; 14.1 Semisimple elements; 14.2 Connectedness of centralizers; 15 Representations of algebraic groups; 15.1 Weight theory; 15.2 Irreducible highest weight modules.
- 16 Representation theory and maximal subgroups16.1 Dual modules and restrictions to Levi subgroups; 16.2 Steinberg's tensor product theorem; 17 Structure of parabolic subgroups, II; 17.1 Internal modules; 17.2 The theorem of Borel and Tits; 18 Maximal subgroups of classical type simple algebraic groups; 18.1 A reduction theorem; 18.2 Maximal subgroups of the classical algebraic groups; 19 Maximal subgroups of exceptional type algebraic groups; 19.1 Statement of the result; 19.2 Indications on the proof; 20 Exercises for Part II; PART III FINITE GROUPS OF LIE TYPE; 21 Steinberg endomorphisms.
- 21.1 Endomorphisms of linear algebraic groups21.2 The theorem of Lang-Steinberg; 22 Classification of finite groups of Lie type; 22.1 Steinberg endomorphisms; 22.2 The finite groups; 23 Weyl group, root system and root subgroups; 23.1 The root system; 23.2 Root subgroups; 24 A BN-pair for GF; 24.1 Bruhat decomposition and the order formula; 24.2 BN-pair, simplicity and automorphisms; 25 Tori and Sylow subgroups; 25.1 F-stable tori; 25.2 Sylow subgroups; 26 Subgroups of maximal rank; 26.1 Parabolic subgroups and Levi subgroups; 26.2 Semisimple conjugacy classes.