线性代数群第2版

　　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.

暂缺《线性代数群第2版》作者简介

Introduction to the First Edition
Introduction to the Second Edition
Conventions and Notation
CHAPTER 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
　References
CHAPTER 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 Decomposition
CHAPTER 11 Homogeneous Spaces
　5. Semi-lnvariants
　6. Homogeneous Spaces
　7. Algebraic Groups in Characteristic Zero
CHAPTER 111 Solvable Groups
　8. Diagonalizable Groups and Tori
　9. Conjugacy Classes and Centralizers of Scmi-Simple Elements
　10. Connected Solvable Groups
CHAPTER 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 Groups
CHAPTER 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 Groups
References for Chapters I to V
Index of Definition
Index of Notation