線性代數群：英文（第二版）

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 toil
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 Borei groups
Notes
7. Weyi 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. Reduetive 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. I. Diagonalizable groups over F
13.2. F-tori
13.3. Tori in F-groups
13.4. The groups P(λ)
Notes
14. Solvable F-groups
14. I. 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-reduetive 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