Abstract

A Lie algebra is a structure we attach to a matrix group, which encodes many of its algebraic properties. The representation theory of Lie algebras has great applications in physics, which I shall not discuss.

The theory of highest weight representations is now one of the most prolific and useful concepts in representation theory. In this talk, I shall recall how the representation of Lie algebras over fields of characteristic zero (the `ordinary’ case) were first classified using highest weight modules.

In the second part of the talk I shall compare the known results in characteristic zero to the more mysterious modular realm, where our Lie algebra is defined over fields of positive characteristic. Here we see that the simple modules have bounded dimension, and a full classification of simple modules in not known in general. I shall try to describe some of the useful concepts involved in studying the theory, and compare to the ordinary case.