Linear Algebraic Groups [Paperback]

This revised, enlarged edition of Linear Algebraic Groups (1969) starts by presenting foundational material on algebraic groups, Lie algebras, transformation spaces, and quotient spaces. It then turns to solvable groups, general properties of linear algebraic groups, and Chevallys structure theory of reductive groups over algebraically closed groundfields. It closes with a focus on rationality questions over non-algebraically closed fields.

AGBackground 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. Splittló²