Quantum groups and their representations, math 261B, Spring 2009 - N. Reshetikhin


office: 917 Evans Hall
email: reshetik@math.berkeley.edu
Seminars: MWF 10-11am, 05 Evans Hall
Office Hours: Monday 4-5:30 pm,

Description


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.

Suggested References


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.

Tentative syllabus


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.

Lecture 1

After short introduction Lie bialgebras were defined. The meaning of Lie bialgebras in terms of Chevalley complex for Lie algebras.

Lecture 2

A crush course in Poisson and symplectic geometry.

Lecture 3

Poisson Lie groups.

Lecture 4

Tangent Lie bialgebra.Representations of braid groups in the tensor product.

Lecture 5

Quansitriangluar Lie bialgebras. Factorizable Lie bialgebras and Poisson Lie groups.

Lecture 6

The standard Lie bialgebra structure on sl_2 and its dual.

Lecture 7

The standard Poisson Lie structure on SL_2

Lecture 8

The dual Poisson Lie group to SL_2

Lecture 9

The double construction for Lie bialgebras.

Lecture 10

The double construction for Lie bialgebras, the double is a factorizable Li ebialgebra, the double of b in sl_2.

Lecture 11

The double of a Borel subalgebra in a simple Lie algebra with the standard Lie bialgebra structure. Corresponding r-matrices for simple Lie bialgebras.

Lecture 12

No lecture

Lecture 13

Real forms of Lie bialgebras and Poisson Lie groups. Poisson Lie actions. The double of a Poisson Lie group and its properties.

Lecture 14

Symplectic leaves of Poisson Lie groups.

Lecture 15

Lecture 16

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