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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

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.