I. Algebraic Geometry. - 0. Some Commutative Algebra. - 1. Affine and Projective Varieties. - 2. Varieties. - 3. Dimension. - 4. Morphisms. - 5. Tangent Spaces. - 6. Complete Varieties. - II. Affine Algebraic Groups. - 7. Basic Concepts and Examples. - 8. Actions of Algebraic Groups on Varieties. - III. Lie Algebras. - 9. Lie Algebra of an Algebraic Group. - 10. Differentiation. - IV. Homogeneous Spaces. - 11. Construction of Certain Representations. - 12. Quotients. - V. Characteristic 0 Theory. - 13. Correspondence between Groups and Lie Algebras. - 14. Semisimple Groups. - VI. Semisimple and Unipotent Elements. - 15. Jordan-Chevalley Decomposition. - 16. Diagonalizable Groups. - VII. Solvable Groups. - 17. Nilpotent and Solvable Groups. - 18. Semisimple Elements. - 19. Connected Solvable Groups. - 20. One Dimensional Groups. - VIII. Borel Subgroups. - 21. Fixed Point and Conjugacy Theorems. - 22. Density and Connectedness Theorems. - 23. Normalizer Theorem. - IX. Centralizers of Tori. - 24. Regular and Singular Tori. - 25. Action of a Maximal Torus on G/?. - 26. The Unipotent Radical. - X. Structure of Reductive Groups. - 27. The Root System. - 28. Bruhat Decomposition. - 29. Tits Systems. - 30. Parabolic Subgroups. - XI. Representations and Classification of Semisimple Groups. - 31. Representations. - 32. Isomorphism Theorem. - 33. Root Systems of Rank 2. - XII. Survey of Rationality Properties. - 34. Fields of Definition. - 35. Special Cases. - Appendix. Root Systems. - Index of Terminology. - Index of Symbols. Language: English