Appendix: a review of algebraic groups

Summary

Inevitably in this text, we have needed to use ideas from the theory of semisimple and reductive algebraic groups, though in practice we have been mostly concerned with these ideas as they pertain to GL_n or SL_n. There is an ample literature on algebraic group theory: proofs of all the assertions below may be found either in the standard texts of Humphreys [1987] or Jantzen [1987, II 1, 2]. I urge you to read this Appendix (if at all) only after digesting Chapter 1. We assume throughout that K is algebraically closed.

Linear algebraic groups: definitions

Let us view the polynomial ring K[x₁,…,x_n] as the polynomial functions on an n-dimensional space, Aⁿ(K), over K. Regard Aⁿ(K) as an affine space, as opposed to a vector space, since, given any point a = (a₁,…,a_n) in it we cannot distinguish K[x₁,…,x_n] from K[x₁ – a₁,…,x_n – a_n] (so a becomes the origin with this new viewpoint). By an affine variety we mean a collection of points in Aⁿ(K) given by the simultaneous vanishing of a set of polynomials.

Definition 1 A linear algebraic groupG over K is an affine variety V furnished with a compatible group structure. This means that multiplication V × V → V and inversion V → V are morphisms of varieties (which is to say that the coordinates of the map are given by polynomials). Denote the affine algebra or coordinate ring of V by K[V].