Linear algebraic groups N. Perrin November 9, 2015 2 Contents 1 First definitions and properties 7 1.1 Algebraic groups . .7 1.1.1 Definitions . .7 1.1.2 Chevalley's Theorem . .7 1.1.3 Hopf algebras . .8 1.1.4 Examples . .8 1.2 First properties . 10 1.2.1 Connected components . 10 1.2.2 Image of a group homomorphism . 10 1.2.3 Subgroup generated by subvarieties . 11 1.3 Action on a variety . 12 1.3.1 Definition . 12 1.3.2 First properties . 12 1.3.3 Affine algebraic groups are linear . 14 2 Tangent spaces and Lie algebras 15 2.1 Derivations and tangent spaces . 15 2.1.1 Derivations . 15 2.1.2 Tangent spaces . 16 2.1.3 Distributions . 18 2.2 Lie algebra of an algebraic group . 18 2.2.1 Lie algebra . 18 2.2.2 Invariant derivations . 19 2.2.3 The distribution algebra . 20 2.2.4 Envelopping algebra . 22 2.2.5 Examples . 22 2.3 Derived action on a representation . 23 2.3.1 Derived action . 23 2.3.2 Stabilisor of the ideal of a closed subgroup . 24 2.3.3 Adjoint actions . 25 3 Semisimple and unipotent elements 29 3.1 Jordan decomposition . 29 3.1.1 Jordan decomposition in GL(V ).......................... 29 3.1.2 Jordan decomposition in G ............................. 30 3.2 Semisimple, unipotent and nilpotent elements . 31 3.3 Commutative groups . 32 3.3.1 Diagonalisable groups . 32 3 4 CONTENTS 3.3.2 Structure of commutative groups . 33 4 Diagonalisable groups and Tori 35 4.1 Structure theorem for diagonalisable groups . 35 4.1.1 Characters . 35 4.1.2 Structure Theorem . 36 4.2 Rigidity of diagonalisable groups . 38 4.3 Some properties of tori . 39 4.3.1 Centraliser of Tori . 39 4.3.2 Pairing . 39 5 Unipotent and sovable groups 41 5.1 Definitions . 41 5.1.1 Groups . 41 5.1.2 Lie algebras . 42 5.1.3 Upper triangular matrices . 42 5.2 Lie-Kolchin Theorems . 42 5.2.1 Burnside and Wederburn Theorem . 42 5.2.2 Unipotent groups . 43 5.2.3 Solvable groups . 43 5.3 Structure Theorem . 44 5.3.1 Statement of the existence of quotients . 44 5.3.2 Structure Theorem . 45 6 Quotients 51 6.1 Differentials . 51 6.1.1 Module of K¨ahlerdifferentials . 51 6.1.2 Back to tangent spaces . 54 6.2 Separable morphisms . 55 6.2.1 Separable and separably generated extensions . 55 6.2.2 Smooth and normal varieties . 58 6.2.3 Separable and birational morphisms . 58 6.2.4 Application to homogeneous spaces . 61 6.2.5 Flat morphisms . 62 6.3 Quotients . 62 6.3.1 Chevalley's semiinvariants . 62 7 Borel subgroups 67 7.1 Borel fixed point Theorem . 67 7.1.1 Reminder on complete varieties . 67 7.1.2 Borel fixed point Theorem . 68 7.2 Cartan subgroups . 69 7.2.1 Borel pairs . 69 7.2.2 Centraliser of Tori, Cartan subgroups . 70 7.2.3 Cartan subgroups . 71 7.3 Normalisers of Borel subgroups . 74 7.4 Reductive and semisimple algebraic groups . 74 7.4.1 Radical and unipotent radical . 74 7.4.2 Reductive and semisimple algebraic groups . 75 CONTENTS 5 8 Geometry of the variety of Borel subgroups 77 8.1 The variety of Borel subgroups . 77 8.2 Action of a torus on a projective space . 79 8.3 Cartan subgroups of a reductive group . 80 9 Structure of reductive groups 85 9.1 First definitions and results . 85 9.1.1 Examples . 85 9.1.2 Root datum . 85 9.2 Centraliser of semisimple elements . 87 9.3 Structure theorem for reductive groups . 88 9.4 Semisimple groups of rank one . 89 9.4.1 Rank one and PGL2 ................................. 89 9.4.2 Groups of semisimple rank one . 91 9.5 Structure Theorem . 93 9.5.1 Root datum of a reductive group . 93 9.5.2 Weyl group . 95 9.5.3 Subgroups normalised by T ............................. 97 9.5.4 Bialynicki-Birula decomposition and Bruhat decomposition . 100 9.6 Structure of semisimple groups . 104 10 Representations of semisimple algebraic groups 107 10.1 Basics on representations . 107 10.2 Parabolic subgroups of G .................................. 109 10.2.1 Existence of maximal parabolic subgroups . 109 10.2.2 Description of all parabolic subgroups . 110 10.3 Existence of representations . 112 11 Uniqueness and existence Theorems, a review 115 11.1 Uniqueness Theorem . 115 11.1.1 Structure constants . 115 11.1.2 The elements nα ................................... 116 11.1.3 Presentation of G ................................... 117 11.1.4 Uniqueness of structure constants ..