线性代数群
2009-8
世界图书出版公司
A.Borel
288
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Apart from some knowledge of Lie algebras， the main prerequisite for these Notes is some familiarity with algebraic geometry. In fact， comparatively little is actually needed. Most of the notions and results frequently used in the Notes are summarized， a few with proofs， in a preliminary Chapter AG. As a basic reference， we take Mumfords Notes [14]， and have tried to be to some extent self-contained from there. A few further results from algebraic geometry needed on some specific occasions will be recalled （with references） where used. The point of view adopted here is essentially the set theoretic one： varieties are identified with their set of points over an algebraic closure of the groundfield （endowed with the Zariski-topology）， however with some traces of the scheme point of view here and there.
Introduction to the First Edition Introduction to the Second EditionConventions and NotationCHAPTER AG--Background Material From Algebraic Geometry 1. Some Topological Notions 2. Some Facts from Field Theory 3. Some Commutative Algebra 4. Sheaves 5. Affine K-Schemes, Prevarieties 6. Products; Varieties 7. Projective and Complete Varieties 8. Rational Functions; Dominant Morphisms 9. Dimension 10. Images and Fibres of a Morphism 11. k-structures on K-Schemes 12. k-Structures on Varieties 13. Separable points 14. Galois Criteria for Rationality 15. Derivations and Differentials 16. Tangent Spaces 17. Simple Points 18. Normal Varieties ReferencesCHAPTER I--General Notions Associated With Algebraic Groups 1. The Notion of an Algebraic Groups 2. Group Closure; Solvable and Nilpotent Groups 3. The Lie Algebra of an Algebraic Group 4. Jordan DecompositionCHAPTER 11 Homogeneous Spaces 5. Semi-lnvariants 6. Homogeneous Spaces 7. Algebraic Groups in Characteristic ZeroCHAPTER 111 Solvable Groups 8. Diagonalizable Groups and Tori 9. Conjugacy Classes and Centralizers of Scmi-Simple Elements 10. Connected Solvable GroupsCHAPTER IV -- Borel Subgroups; Rcductive Groups 11. Borei Subgroups 12. Caftan Subgroups; Regular Elements 13. The Borel Subgroups Containing a Given Torus 14. Root Systems and Bruhat Decomposition in Reductive GroupsCHAPTER V-- Rationality Questions 15. Split Solvable Groups and Subgroups 16. Groups over Finite Fields 17. Quotient of a Group by a Lie Subalgebra 18. Cartan Subgroups over the Groundfield. Unirationality. Splitting of Reductive Groups 19. Cartan Subgroups of Solvable Groups 20. lsotropic Reductive Groups 21. Relative Root System and Bruhat Decomposition for lsotropic ReductiveGroups 22. Central lsogenies 23. Examples 24. Survey of Some Other Topics A. Classification B. Linear Representations C. Real Reductive GroupsReferences for Chapters I to VIndex of DefinitionIndex of Notation
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