線性代數(shù)群

內(nèi)容概要

　　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.

書籍目錄

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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