## 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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