Math 735 — Spring 2002 Topics in Algebra: Quantum Groups

Various applications of Hopf algebras were recently found in the fields of mathematical physics, topology, combinatorics, ring theory, completely integrable systems, and others. They extended the role of Lie groups as symmetries of different mathematical structures. We plan to discuss some of these applications according to the audience interests:

• Structure of Hopf algebras
• Representations of quantized enveloping algebras of semisimple and affine Lie algebras
• Related semiclassical structures - Poisson-Lie groups
• Invariants of knots via Hopf algebras
• Canonical bases and other combinatorial structures
• Applications to mathematical physics, Knizhnik-Zamolodchikov equation
• Completely integrable systems

There are no prerequisites for the course except knowing some elementary facts on semisimple Lie algebras and groups. We will establish connections with the topics of the courses of Professors Brown and Barbasch (Spring and Fall of 2001, respectively) but will not overlap with them.

Textbooks:

A Guide to Quantum Groups, V. Chary and A. Pressley, Cambridge Univ. Press, 1994.

Lectures on Representation Theory and Knizhnik-Zamolodchikov Equation, P. Etingof, I Frenkel, A. Kirillov Jr., AMS, 1998.

Hopf Algebras and their Actions on Rings, S. Montgomery, AMS, 1993.