Jie Du (University of New South Wales)

Friday 7th April, 12.05-12.55pm, Carslaw 159

Frobenius morphisms and representations of algebras

Frobenius morphisms are fundamental in the theory of algebraic groups and their representations. It is the key idea to investigate the rational structure of algebraic groups and to relate finite groups of Lie type to the corresponding algebraic groups through certain sophisticated geometric constructions.

In this talk, I will introduce Frobenius morphisms for finite dimensional k-algebras A (not necessarily associative), where k is the algebraic closure of a finite field, and establish a direct link between representations of A and its fixed-point algebra. When applying the theory to associative algebras, especially to quivers with automorphisms, we see that many fundamental properties such as heredity, representation type, global dimension, the Auslander-Reiten theory and so on for an algebra over a finite field are completely determined by the corresponding ones for the algebra over the algebraic closure of the finite field. Thus, we obtain further applications to Kac theories, derived categories and Ringel-Hall algebras and quantum groups.

I will also mention a parallel theory for finite dimensional Lie algebras. (This is a joint work with Bangming Deng.)