Lecturer: Henning Haahr Andersen
Content:
Let L be a semisimple complex Lie algebra. In this course we shall begin by studying the class of socalled highest weight modules for L. It contains all finite dimensional representations of L. We shall classify all irreducible modules in this class and for the finite dimensional ones also determine their characters and dimensions. This we do by proving the famous Weyl character formula.
Next we shall introduce the quantum group associated to L. We construct it via generators and relations in a way resembling Serre's theorem for semisimple Lie algebras. So when defined like this a quantum group is not a group but an algebra. We shall see that it has a Hopf algebra structure and that its representation theory is very similar to that of L.
The course will thus contain classical theory as well as results found during the last decade. A couple of examples will take us right to the current research frontier.
Prerequisites: Lie algebra theory corresponding to Representation Theory Fall00 (or Humphreys' book, I-V)
Literature: J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, VI Informal notes.