Linear Algebraic GroupsJames E. Humphreys is presently Professor of Mathematics at the University of Massachusetts at Amherst. Before this, he held the posts of Assistant Professor of Mathematics at the University of Oregon and Associate Professor of Mathematics at New York University. His main research interests include group theory and Lie algebras. He graduated from Oberlin College in 1961. He did graduate work in philosophy and mathematics at Cornell University and later received hi Ph.D. from Yale University if 1966. In 1972, Springer-Verlag published his first book, "Introduction to Lie Algebras and Representation Theory" (graduate Texts in Mathematics Vol. 9). |
Contents
| 1 | |
Basic Concepts and Examples | 7 |
Quotients | 12 |
Varieties | 16 |
Dimension | 24 |
Tangent Spaces | 37 |
Actions of Algebraic Groups on Varieties | 58 |
Lie Algebras | 65 |
Borel Subgroups | 133 |
Normalizer Theorem | 143 |
Action of a Maximal Torus on GB | 151 |
The Unipotent Radical | 157 |
Structure of Reductive Groups | 163 |
Bruhat Decomposition | 169 |
Tits Systems | 175 |
Parabolic Subgroups | 183 |
Differentiation | 71 |
Homogeneous Spaces | 79 |
Semisimple Groups | 90 |
Diagonalizable Groups | 101 |
Solvable Groups | 109 |
Table of Contents | 115 |
Connected Solvable Groups | 121 |
Table of Contents | 193 |
Root Systems of Rank 2 | 207 |
Survey of Rationality Properties | 217 |
Appendix Root Systems | 229 |
| 241 | |
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Common terms and phrases
affine algebra affine open sets affine open subset affine variety algebraic group arbitrary automorphism bijective Borel subgroup CG(S char Chevalley closed sets closed subgroup closed subset codimension commutative conjugate connected constructible containing Corollary corresponding cosets d-group defined denote dimension Exercise finite dimensional follows G acts G₁ GL(n GL(V group G hence homogeneous homomorphism implies induces intersection irreducible affine variety irreducible components irreducible variety isomorphism K-algebra Lemma Let G Lie algebra linear matrices maximal torus morphism of algebraic multiplication nilpotent nonempty nonzero normal subgroup open sets open subset parabolic subgroup prevarieties prime ideals projective varieties Proof Proposition prove representation resp ring root system semisimple elements shows SL(n solvable subgroup of G subspace surjective T₁ tangent space Theorem topology unipotent unique v₁ vanishing vector space Weyl group xe G Zariski Zariski topology