| 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 |
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