Lie Groups and Representations:
Mathematics G4344 (Spring 2016)
Monday and Wednesday 1:10-2:25
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
(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
There will be problem sets due roughly each week.
Set 1. Due Monday February 1.
Set 2. Due Monday February 8.
Set 3. Due Monday February 15.
Set 4. Due Monday February 22.
Set 5. Due Monday February 29.
Set 6. Due Monday March 7.
Set 7. Due Monday March 21.
Set 8. Due Monday April 4.
Set 9. Due Monday April 11.
Set 10. Due Monday April 18.
Problem Set. Due Wednesday May 4.
Alexander Kirillov, Jr.
An Introduction to Lie Groups and
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
For the first section of the class, other references are
Serre, Complex Semisimple Lie Algebras
Knapp, Lie Groups: Beyond an Introduction
The following selection of on-line lecture notes and course
materials may be useful:
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.