Author(s): Tom de Medts
Series: lecture notes
Edition: version 2019-02-25
Year: 2019
Language: English
Commentary: Downloaded from https://algebra.ugent.be/~tdemedts/files/LinearAlgebraicGroups-TomDeMedts.pdf
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
