斯普林格所著的《线性代数群(第2版)》的第一版讲述了代数封闭域上的线性代数群表示理论,这是第二版,做了全面的修订,扩充,将此理论扩充到任意域上,即不一定需要代数封闭。本书在理论上也上了一个层次,但风格上和第一版保持一致,自成体系,将所需的代数几何和交换代数部分,以及基本约化群基本知识囊括其中。
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书名 | 线性代数群(第2版) |
分类 | 科学技术-自然科学-数学 |
作者 | (荷)斯普林格 |
出版社 | 世界图书出版公司 |
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简介 | 编辑推荐 斯普林格所著的《线性代数群(第2版)》的第一版讲述了代数封闭域上的线性代数群表示理论,这是第二版,做了全面的修订,扩充,将此理论扩充到任意域上,即不一定需要代数封闭。本书在理论上也上了一个层次,但风格上和第一版保持一致,自成体系,将所需的代数几何和交换代数部分,以及基本约化群基本知识囊括其中。 目录 Preface to the Second Edition 1. Some Algebraic Geometry 1.1. The Zariski topology 1.2. Irreducibility of topological spaces 1.3. Affine algebras 1.4. Regular functions, ringed spaces 1.5. Products 1.6. Prevarieties and varieties 1.7. Projective varieties 1.8. Dimension 1.9. Some results on morphisms Notes 2. Linear Algebraic Groups, First Properties. 2.1. Algebraic groups 2.2. Some basic results 2.3. G-spaces 2.4. Jordan decomposition 2.5. Recovering a group from its representations Notes 3. Commutative Algebraic Groups 3.1. Structure of commutative algebraic groups 3.2. Diagonalizable groups and tori 3.3. Additive functions 3.4. Elementary unipotent groups Notes 4. Derivations, Differentials, Lie'Algebras 4.1. Derivations and tangent spaces 4.2. Differentials, separability 4.3. Simple points 4.4. The Lie algebra of a linear algebraic group Notes 5. Topological Properties of Morphisms, Applications 5.1. Topological properties of morphisms 5.2. Finite morphisms, normality 5.3. Homogeneous spaces 5.4. Semi-simple automorphisms 5.5. Quotients Notes 6. Parabolic Subgroups, Borel Subgroups, Solvable Groups 6.1. Complete varieties 6.2. Parabolic subgroups and Borel subgroups 6.3. Connected solvable groups 6.4. Maximal tori, further properties of Borel groups Notes 7. Weyl Group, Roots, Root Datum 7.1. The Weyl group 7.2. Semi-simple groups of rank one 7.3. Reductive groups of semi-simple rank one 7.4. Root data 7.5. Two roots 7.6. The unipotent radical Notes 8. Reductive Groups 8.1. Structural properties of a reductive group 8.2. Borel subgroups and systems of positive roots 8.3. The Bruhat decomposition 8.4. Parabolic subgroups 8.5. Geometric questions related to the Bruhat decomposition Notes 9. The Isomorphism Theorem 9.1. Two dimensional root systems 9.2. The structure constants 9.3. The elements nα 9.4. A presentation of G 9.5. Uniqueness of structure constants 9.6. The isomorphism theorem Notes 10. The Existence Theorem 10.1. Statement of the theorem, reduction 10.2. Simply laced root systems 10.3. Automorphisms, end of the proof of 10.1.1 Notes 11. More Algebraic Geometry 11.1. F-structures on vector spaces 11.2. F-varieties: density, criteria for ground fields 11.3. Forms 11.4. Restriction of the ground field Notes 12. F-groups: General Results 12.1. Field of definition of subgroups 12.2. Complements on quotients 12.3. Galois cohomology 12.4. Restriction of the ground field Notes 13. F-tori 13.1. Diagonalizable groups over F 13.2. F-toil 13.3. Toil in F-groups 13.4. The groups P(λ) Notes 14. Solvable F-groups 14.1. Generalities 14.2. Action of Ga on an affine variety, applications 14.3. F-split solvable groups 14.4. Structural properties of solvable groups Notes 15. F-reductive Groups 15.1. Pseudo-parabolic F-subgroups 15.2. A fixed point theorem 15.3. The root datum of an F-reductive group 15.4. The groups U(a) 15.5. The index Notes 16. Reduetive F-groups 16.1. Parabolic subgroups 16.2. Indexed root data 16.3. F-split groups 16.4. The isomorphism theorem 16.5. Existence Notes 17. Classification 17.1. Type An-1 17.2. Types Bn and Cn 17.3. Type Dn 17.4. Exceptional groups, type G2 17.5. Indices for types F4 and E8 17.6. Descriptions for type F4 17.7. Type E6 17.8. Type E7 17.9. Trialitarian type D4 17.10. Special fields Notes Table of Indices Bibliography Index |
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