Lie Groups and Representations:
Mathematics G4344 (Spring 2016)

Monday and Wednesday 1:10-2:25

507 Mathematics

This course will cover various more advanced aspects of the theory
of Lie groups, Lie algebras and their representations. It is
aimed at first-year mathematics graduate students although students
in physics might also find it of interest. It is a
continuation of the fall semester course taught by Andrei Okounkov.

Topics to be covered will include some of the following:

Classification of complex semi-simple Lie algebras

Verma modules and highest-weight representations

Harish-Chandra homomorphism

(for these topics a reference will be the Kirillov textbook)

Lie algebra cohomology and the Borel-Weil-Bott theorem (algebraic)

Borel-Weil theory, algebraic geometry methods

Symplectic geometry and the orbit method

Heisenberg group, Stone-von Neumann theorem, some quantum mechanics

The metaplectic representation

Clifford algebras, spin groups and the spinor representation

SL(2,R) and its representations

Representation theory and modular forms

### Problem Sets

There will be problem sets due roughly each week.

Problem
Set 1. Due Monday February 1.

Problem
Set 2. Due Monday February 8.

Problem
Set 3. Due Monday February 15.

Problem
Set 4. Due Monday February 22.

Problem
Set 5. Due Monday February 29.

Problem
Set 6. Due Monday March 7.

Problem
Set 7. Due Monday March 21.

Problem
Set 8. Due Monday April 4.

Problem
Set 9. Due Monday April 11.

Problem
Set 10. Due Monday April 18.

Final
Problem Set. Due Wednesday May 4.

### Textbook

Alexander Kirillov, Jr.

An Introduction to Lie Groups and
Lie Algebras

Cambridge University Press, 2008

Note that electronic version of this book is available freely for
Columbia students at the link above or via its entry in the Columbia
library catalog.

For the first section of the class, other references are

Serre, Complex Semisimple Lie Algebras

Knapp, Lie Groups: Beyond an Introduction

Online Resources

The following selection of on-line lecture notes and course
materials may be useful:

Berkeley
Lectures on Lie Groups and Quantum Groups

David Ben-Zvi course on representations of SL2, see notes
on this page.

Eckhard Meinrenken lecture notes on Lie
Groups and Lie Algebras.