Description
Book SynopsisAGBackground 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.- IGeneral 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.- II Homogeneous Spaces.- 5. Semi-Invariants.- 6. Homogeneous Spaces.- 7. Algebraic Groups in Characteristic Zero.- III Solvable Groups.- 8. Diagonalizable Groups and Tori.- 9. Conjugacy Classes and Centralizers of Semi-Simple Elements.- 10. Connected Solvable Groups.- IVBorel 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.- VRationality 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. Isotropic Reductive Groups.- 21. Relative Root System and Bruhat Decomposition for Isotropic Reductive Groups.- 22. Central Isogenies.- 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.
Table of ContentsAG—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.- 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.- II — Homogeneous Spaces.- §5. Semi-Invariants.- §6. Homogeneous Spaces.- §7. Algebraic Groups in Characteristic Zero.- III Solvable Groups.- §8. Diagonalizable Groups and Tori.- §9. Conjugacy Classes and Centralizers of Semi-Simple Elements.- §10. Connected Solvable Groups.- 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.- 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. Isotropic Reductive Groups.- §21. Relative Root System and Bruhat Decomposition for Isotropic Reductive Groups.- §22. Central Isogenies.- §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.
