Summary: Linear Algebraic Groups: a Crash Course
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.
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
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.