Quantum groups and their representations, math 261B, Spring 2009 - N. Reshetikhin
office: 917 Evans Hall
Seminars: MWF 10-11am, 05 Evans Hall
Office Hours: Monday 4-5:30 pm,
The goal of this part of the Lie groups and Lie algebras course is an introduction to quantum groups. The subject was developed over the last 20 years. Many ideas and techniques are natural extensions of those in Lie groups, Lie algebras, and their representation theory. The main hero of this course (at least in the first part of it) will be quantum $sl_2$. Elements of symplectic geometry will be used in parts of this course. A symplectic geometry course as a pre-requisite is desirable, but not necessary.
There will be home-works and at the end of he course a take-home final exam.
Lecture notes will appear as the course will progress. The links to the lecture notes are on the dates of lectures.
1. V. Chari, A. Pressley. , "A guide to quantum groups", Cambridge University Press, 1994.
2. P. Etingof and O. Schiffman. Lectures on Quantum Groups. International Press, 1998.
3. C. Kassel. Quantum Groups (Springer: 1994). ISBN 0-387-94370-6
4. L. Korogodsky, Y. Soibelman, Alegbras of functions on quantum groups. I, Amer. Math. Soc., 1997.
The syllabus will evolve as the course will go on. Here is a tentative (perhaps too ambitious) syllabus. Corrections will be posted. It is very likely that the course will go much slower then outlined.
After short introduction Lie bialgebras were defined. The meaning of Lie bialgebras in terms of Chevalley complex for Lie algebras.
A crush course in Poisson and symplectic geometry.
Poisson Lie groups.
Tangent Lie bialgebra.Representations of braid groups in the tensor product.
Quansitriangluar Lie bialgebras. Factorizable Lie bialgebras and Poisson Lie groups.
The standard Lie bialgebra structure on sl_2 and its dual.
The standard Poisson Lie structure on SL_2
The dual Poisson Lie group to SL_2
The double construction for Lie bialgebras.
The double construction for Lie bialgebras, the double is a factorizable Li ebialgebra, the double of b in sl_2.
The double of a Borel subalgebra in a simple Lie algebra with the standard Lie bialgebra structure. Corresponding r-matrices for simple Lie bialgebras.
Real forms of Lie bialgebras and Poisson Lie groups. Poisson Lie actions. The double of a Poisson Lie group and its properties.
Symplectic leaves of Poisson Lie groups.