9 - Tilting modules for algebraic groups and finite dimensional algebras

Summary

Introduction

This is a survey article describing some of the ways in which the theory of rational representations of algebraic groups interacts with the representation theory of finite dimension algebras, with particular emphasis on tilting modules.

In its simplest form the connection between the two areas is the following. Let G be a linear algebraic group over an algebraically closed field k. Then the coordinate algebra k[G] is naturally a commutative Hopf algebra, in particular a coalgebra. A coalgebra is the union of its finite dimensional subcoalgebras and, for a finite dimensional coalgebra C, say, there is a natural equivalence of categories between the category of C-comodules and the category of modules for the dual algebra C*. A (rational) G-module is, more or less by definition, a k[G] – comodule. If V is a finite dimensional right comodule, with structure map τ: V → V ⊗ k [G], then the image of τ lies in V ⊗ C, for some finite dimensional subcoalgebra C of k[G] and V is naturally a right C-comodule and hence a left C*-module. Thus the (finite dimensional) representation theory of G is simply the union of the (finite dimensional) representation theories of the finite dimensional algebras C*, as C ranges over finite dimensional subcoalgebras of k[G].

Of course all of this is so far too general to be of any particular use.