Linear Algebraic Groups: a Crash Course Dave Anderson Summary: Linear Algebraic Groups: a Crash Course Dave Anderson January 24, 2011 This is a collection of notes for three lectures, designed to introduce linear algebraic groups quickly in a course on Geometric Invariant Theory. There are several good introductory textbooks; in particular, the books by Humphreys [H], Springer [S], and Borel [B]. Here I merely distill some of the material from Humphreys and Springer. 1 Definitions We'll work over a fixed algebraically closed base field k. Definition 1.1 An algebraic group G is a group object in the category of varieties over k. That is, G is a group and a variety, and the maps G × G G and G G (g, h) gh g g-1 are morphisms of varieties. (And there is a distinguished k-point e G, the identity.) A homomorphism of algebraic groups is a group homomorphism that is also a map of varieties. In schemey language, another way to say this is that the functor hG : Schemes Sets factors through Groups. Collections: Mathematics