Quantum groups
3-4 hours of lectures per week.
Lecturer Henning Haahr Andersen
Contents Quantum groups were introduced independently some 15 years ago
(independently by a Russian and a Japanese school of mathematicians). It
was soon realized that these objects could be useful in many areas of modern mathematics.
In representation theory they turned out to be very well suited for dealing with
representations of semisimple Lie algebras and algebraic groups. They
have given rise to completely new results and have at the same time provided the key ingredients
in solution of older problems.
In this course we shall construct them via generators and relations. As we shall discover a quantum
group is not a group but an algebra. We shall see that it has a Hopf algebra structure and that many
aspects of its representation theory are very similar to that of the corresponding Lie algebra or Lie group.
In particular we shall develop its highest weight theory thereby classifying all finite dimensional representations.
The course will 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 Algebra II (or similar)
Text-books Informal notes. Articles.
(To get an impression of the many relations between quantum groups
and various other mathematical subjects see e.g. the book by V. Chari
and A. Pressley "A guide to quantum groups".)