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.


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.