Linear algebraic groups线性代数群
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作者： Armand Borel著
出 版 社：
出版时间： 1991-4-1字数：版次： 1页数： 288印刷时间： 1991/04/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387973708包装： 精装内容简介
This book is a revised and enlarged edition of "Linear Algebraic Groups", published by W.A. Benjamin in 1969. The text of the first edition has been corrected and revised. Accordingly, this book presents foundational material on algebraic groups, Lie algebras, transformation spaces, and quotient spaces. After establishing these basic topics, the text then turns to solvable groups, general properties of linear algebraic groups and Chevally's structure theory of reductive groups over algebraically closed groundfields. The remainder of the book is devoted to rationality questions over non-algebraically closed fields. This second edition has been expanded to include material on central isogenies and the structure of the group of rational points of an isotropic reductive group. The main prerequisite is some familiarity with algebraic geometry. The main notions and results needed are summarized in a chapter with references and brief proofs.
目录
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 II Homogeneous Spaces
5. Semi-Invariants
6. Homogeneous Spaces
7. Algebraic Groups in Characteristic Zero
CHAPTER III Solvable Groups
8. Diagonalizable Groups and Tori
9. Conjugacy Classes and Centralizers of Semi-Simple Elements
10. Connected Solvable Groups
CHAPTER IV -Borel Subgroups; Reductive Groups
11. Borel Subgroups
12. Cartan 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 Reductive Groups
22. Central lsogenies
23. Examples
24. Survey of Some Other Topics
A. Classification
B. Linear Representations
C. Real Reductive Groups
References for Chapters 1 to V
Index of Definition
Index of Notation
