Reference for Algebraic Groups in Ergodic Theory

Question

It seems that the theory of algebraic groups is used in ergodic theory. I was hoping someone could recommend an introduction to algebraic groups that

1. assumes a knowledge of commutative algebra

2. covers or begins to cover those aspects of algebraic groups which are useful in dynamics and ergodic theory

3. doesn't venture too far into abstractions that are unnecessary from the viewpoint of applications to dynamics.

Thank you.

Answer 1

One thing to note is that the algebraic groups that come up in e.g. Ratner's theorem, or the recent work of Eskin--Mirzakhani, are affine algebraic groups. These can be studied fairly concretely, and need less algebraic geometry than many other topics (for example, because at leasts for the basics you don't need to get into the study of projective varieties.)

Have you looked at Dave Witte Morris's book on Ratner's theorem? It summarizes some basics in a pretty succinct manner.

Answer 2

A classical reference in this topic is Robert Zimmer's book Ergodic Theory and Semisimple Groups. In his notes is an short appendix on algebraic groups. There are also many other references about algebraic groups, e.g., the notes of J.S. Milne here.