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。

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