Linear Algebraic Groups (Hardcover, 5)

I. Algebraic Geometry.- 0. Some Commutative Algebra.- 1. Affine and Projective Varieties.- 1.1 Ideals and Affine Varieties.- 1.2 Zariski Topology on Affine Space.- 1.3 Irreducible Components.- 1.4 Products of Affine Varieties.- 1.5 Affine Algebras and Morphisms.- 1.6 Projective Varieties.- 1.7 Products of Projective Varieties.- 1.8 Flag Varieties.- 2. Varieties.- 2.1 Local Rings.- 2.2 Prevarieties.- 2.3 Morphisms.- 2.4 Products.- 2.5 Hausdorff Axiom.- 3. Dimension.- 3.1 Dimension of a Variety.- 3.2 Dimension of a Subvariety.- 3.3 Dimension Theorem.- 3.4 Consequences.- 4. Morphisms.- 4.1 Fibres of a Morphism.- 4.2 Finite Morphisms.- 4.3 Image of a Morphism.- 4.4 Constructible Sets.- 4.5 Open Morphisms.- 4.6 Bijective Morphisms.- 4.7 Birational Morphisms.- 5. Tangent Spaces.- 5.1 Zariski Tangent Space.- 5.2 Existence of Simple Points.- 5.3 Local Ring of a Simple Point.- 5.4 Differential of a Morphism.- 5.5 Differential Criterion for Separability.- 6. Complete Varieties.- 6.1 Basic Properties.- 6.2 Completeness of Projective Varieties.- 6.3 Varieties Isomorphic to P1.- 6.4 Automorphisms of P1.- II. Affine Algebraic Groups.- 7. Basic Concepts and Examples.- 7.1 The Notion of Algebraic Group.- 7.2 Some Classical Groups.- 7.3 Identity Component.- 7.4 Subgroups and Homomorphisms.- 7.5 Generation by Irreducible Subsets.- 7.6 Hopf Algebras.- 8. Actions of Algebraic Groups on Varieties.- 8.1 Group Actions.- 8.2 Actions of Algebraic Groups.- 8.3 Closed Orbits.- 8.4 Semidirect Products.- 8.5 Translation of Functions.- 8.6 Linearization of Affine Groups.- III. Lie Algebras.- 9. Lie Algebra of an Algebraic Group.- 9.1 Lie Algebras and Tangent Spaces.- 9.2 Convolution.- 9.3 Examples.- 9.4 Subgroups and Lie Subalgebras.- 9.5 Dual Numbers.- 10. Differentiation.- 10.1 Some Elementary Formulas.- 10.2 Differential of Right Translation.- 10.3 The Adjoint Representation.- 10.4 Differential of Ad.- 10.5 Commutators.- 10.6 Centralizers.- 10.7 Automorphisms and Derivations.- IV. Homogeneous Spaces.- 11. Construction of Certain Representations.- 11.1 Action on Exterior Powers.- 11.2 A Theorem of Chevalley.- 11.3 Passage to Projective Space.- 11.4 Characters and Semi-Invariants.- 11.5 Normal Subgroups.- 12. Quotients.- 12.1 Universal Mapping Property.- 12.2 Topology of Y.- 12.3 Functions on Y.- 12.4 Complements.- 12.5 Characteristic 0.- V. Characteristic 0 Theory.- 13. Correspondence between Groups and Lie Algebras.- 13.1 The Lattice Correspondence.- 13.2 Invariants and Invariant Subspaces.- 13.3 Normal Subgroups and Ideals.- 13.4 Centers and Centralizers.- 13.5 Semisimple Groups and Lie Algebras.- 14. Semisimple Groups.- 14.1 The Adjoint Representation.- 14.2 Subgroups of a Semisimple Group.- 14.3 Complete Reducibility of Representations.- VI. Semisimple and Unipotent Elements.- 15. Jordan-Chevalley Decomposition.- 15.1 Decomposition of a Single Endomorphism.- 15.2 GL(n, K) and gl(n, K).- 15.3 Jordan Decomposition in Algebraic Groups.- 15.4 Commuting Sets of Endomorphisms.- 15.5 Structure of Commutative Algebraic Groups.- 16. Diagonalizable Groups.- 16.1 Characters and d-Groups.- 16.2 Tori.- 16.3 Rigidity of Diagonalizable Groups.- 16.4 Weights and Roots.- VII. Solvable Groups.- 17. Nilpotent and Solvable Groups.- 17.1 A Group-Theoretic Lemma.- 17.2 Commutator Groups.- 17.3 Solvable Groups.- 17.4 Nilpotent Groups.- 17.5 Unipotent Groups.- 17.6 Lie-Kolchin Theorem.- 18. Semisimple Elements.- 18.1 Global and Infinitesimal Centralizers.- 18.2 Closed Conjugacy Classes.- 18.3 Action of a Semisimple Element on a Unipotent Group.- 18.4 Action of a Diagonalizable Group.- 19. Connected Solvable Groups.- 19.1 An Exact Sequence.- 19.2 The Nilpotent Case.- 19.3 The General Case.- 19.4 Normalizer and Centralizer.- 19.5 Solvable and Unipotent Radicals.- 20. One Dimensional Groups.- 20.1 Commutativity of G.- 20.2 Vector Groups and e-Groups.- 20.3 Properties of p-Polynomials.- 20.4 Automorphisms of Vector Groups.- 20.5 The Main Theorem.- VIII. Borel Subgroups.- 21. Fixed Point and Conjugacy Theorems.- 21.1 Review of Complete Varieties.- 21.2 Fixed Point Theorem.- 21.3 Conjugacy of Borel Subgroups and Maximal Tori.- 21.4 Further Consequences.- 22. Density and Connectedness Theorems.- 22.1 The Main Lemma.- 22.2 Density Theorem.- 22.3 Connectedness Theorem.- 22.4 Borel Subgroups of CG(S).- 22.5 Cartan Subgroups: Summary.- 23. Normalizer Theorem.- 23.1 Statement of the Theorem.- 23.2 Proof of the Theorem.- 23.3 The variety G/B.- 23.4 Summary.- IX. Centralizers of Tori.- 24. Regular and Singular Tori.- 24.1 Weyl Groups.- 24.2 Regular Tori.- 24.3 Singular Tori and Roots.- 24.4 Regular 1-Parameter Subgroups.- 25. Action of a Maximal Torus on G/?.- 25.1 Action of a 1-Parameter Subgroup.- 25.2 Existence of Enough Fixed Points.- 25.3 Groups of Semisimple Rank 1.- 25.4 Weyl Chambers.- 26. The Unipotent Radical.- 26.1 Characterization of Ru(G).- 26.2 Some Consequences.- 26.3 The Groups U?.- X. Structure of Reductive Groups.- 27. The Root System.- 27.1 Abstract Root Systems.- 27.2 The Integrality Axiom.- 27.3 Simple Roots.- 27.4 The Automorphism Group of a Semisimple Group.- 27.5 Simple Components.- 28. Bruhat Decomposition.- 28.1 T-Stable Subgroups of Bu.- 28.2 Groups of Semisimple Rank 1.- 28.3 The Bruhat Decomposition.- 28.4 Normal Form in G.- 28.5 Complements.- 29. Tits Systems.- 29.1 Axioms.- 29.2 Bruhat Decomposition.- 29.3 Parabolic Subgroups.- 29.4 Generators and Relations for W.- 29.5 Normal Subgroups of G.- 30. Parabolic Subgroups.- 30.1 Standard Parabolic Subgroups.- 30.2 Levi Decompositions.- 30.3 Parabolic Subgroups Associated to Certain Unipotent Groups.- 30.4 Maximal Subgroups and Maximal Unipotent Subgroups.- XI. Representations and Classification of Semisimple Groups.- 31. Representations.- 31.1 Weights.- 31.2 Maximal Vectors.- 31.3 Irreducible Representations.- 31.4 Construction of Irreducible Representations.- 31.5 Multiplicities and Minimal Highest Weights.- 31.6 Contragredients and Invariant Bilinear Forms.- 32. Isomorphism Theorem.- 32.1 The Classification Problem.- 32.2 Extension of ?T to N(T).- 32.3 Extension of ?T to Z?.- 32.4 Extension of ?T to TU?.- 32.5 Extension of ?T to ?.- 32.6 Multiplicativity of ?.- 33. Root Systems of Rank 2.- 33.1 Reformulation of (?), (?), (?).- 33.2 Some Preliminaries.- 33.3 Type A2.- 33.4 Type B2.- 33.5 Type G2.- 33.6 The Existence Problem.- XII. Survey of Rationality Properties.- 34. Fields of Definition.- 34.1 Foundations.- 34.2 Review of Earlier Chapters.- 34.3 Tori.- 34.4 Some Basic Theorems.- 34.5 Borel-Tits Structure Theory.- 34.6 An Example: Orthogonal Groups.- 35. Special Cases.- 35.1 Split and Quasisplit Groups.- 35.2 Finite Fields.- 35.3 The Real Field.- 35.4 Local Fields.- 35.5 Classification.- Appendix. Root Systems.- Index of Terminology.- Index of Symbols.