Linear algebraic groups

Preface iii 1 Introduction 1 1.1 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The building bricks . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Finite algebraic groups . . . . . . . . . . . . . . . . . . 3 1.2.2 Abelian varieties . . . . . . . . . . . . . . . . . . . . . 3 1.2.3 Semisimple linear algebraic groups . . . . . . . . . . . 3 1.2.4 Groups of multiplicative type and tori . . . . . . . . . 5 1.2.5 Unipotent groups . . . . . . . . . . . . . . . . . . . . . 5 1.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Solvable groups . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Reductive groups . . . . . . . . . . . . . . . . . . . . . 6 1.3.3 Disconnected groups . . . . . . . . . . . . . . . . . . . 7 1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Algebras 9 2.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . 9 2.2 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Tensor products of K-modules . . . . . . . . . . . . . . 12 2.2.2 Tensor products of K-algebras . . . . . . . . . . . . . . 15 3 Categories 19 3.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . 19 3.2 Functors and natural transformations . . . . . . . . . . . . . . 21 3.3 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . 25 4 Algebraic geometry 31 4.1 Affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 The coordinate ring of an affine variety . . . . . . . . . . . . . 36 4.3 Affine varieties as functors . . . . . . . . . . . . . . . . . . . . 40 5 Linear algebraic groups 43 5.1 Affine algebraic groups . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Closed subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.3 Homomorphisms and quotients . . . . . . . . . . . . . . . . . 54 5.4 Affine algebraic groups are linear . . . . . . . . . . . . . . . . 56 6 Jordan decomposition 63 6.1 Jordan decomposition in GL(V ) . . . . . . . . . . . . . . . . . 63 6.2 Jordan decomposition in linear algebraic groups . . . . . . . . 67 7 Lie algebras and linear algebraic groups 73 7.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.2 The Lie algebra of a linear algebraic group . . . . . . . . . . . 76 8 Topological aspects 83 8.1 Connected components of matrix groups . . . . . . . . . . . . 83 8.2 The spectrum of a ring . . . . . . . . . . . . . . . . . . . . . . 84 8.3 Separable algebras . . . . . . . . . . . . . . . . . . . . . . . . 87 8.4 Connected components of linear algebraic groups . . . . . . . 90 8.5 Dimension and smoothness . . . . . . . . . . . . . . . . . . . . 94 9 Tori and characters 97 9.1 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.2 Diagonalizable groups . . . . . . . . . . . . . . . . . . . . . . . 98 9.3 Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 10 Solvable linear algebraic groups 105 10.1 The derived subgroup of a linear algebraic group . . . . . . . . 105 10.2 The structure of solvable linear algebraic groups . . . . . . . . 107 10.3 Borel subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 113 11 Semisimple and reductive groups 117 11.1 Semisimple and reductive linear algebraic groups . . . . . . . . 117 11.2 The root datum of a reductive group . . . . . . . . . . . . . . 121 11.3 Classification of the root data . . . . . . . . . . . . . . . . . . 131 References 136