Lie Groups and Representations:
Mathematics GR6344 (Spring 2023)
Tuesday and Thursday, 10:10am - 11:25am
507 Mathematics
This course will cover various more advanced aspects of the theory
of Lie groups, Lie algebras and their representations, from a point
of view emphasizing relations to quantum mechanics and number
theory. 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 John Morgan.
Topics to be covered will likely include:
Heisenberg group, Stone-von Neumann theorem, some quantum mechanics
Relation to theta functions
The oscillator representation
Analogy with the spinor representation
Symplectic geometry and the orbit method
Review of universal enveloping algebra, classification theory of
complex Lie algebras
Verma modules and highest-weight representations
Harish-Chandra homomorphism
Lie algebra cohomology and the Borel-Weil-Bott theorem (algebraic)
Classification of finite-dimensional representations of semi-simple
complex Lie algebras
Geometric representation theory. Borel-Weil-Bott theorem
Real semi-simple Lie groups and Lie algebras: classification
SL(2,R) and its representations, relation to modular forms
Schedule
Tuesday, January 17: Overview of course, review of material
from first semester. Notes
Thursday, January 19: The Heisenberg group and its
representations (part 1). Notes
Tuesday, January 24: The Heisenberg group and its
representations (part 2). Notes
Thursday, January 26: More on polarizations. The
symplectic group as automorphisms of the Heisenberg group
Tuesday, January 31: The Poisson bracket. Lie algebra of
the symplectic group. The oscillator representation
Thursday, February 2: Analogy with spin groups and the spinor
representation
Tuesday, February 7: Heisenberg and metaplectic groups over
other fields
Thursday, February 9: Theta functions
Tuesday, February 14: The orbit method and the Heisenberg
group
Thursday, February 16: The orbit method and geometric quantization
in general
Tuesday, February 21:
Thursday, February 23:
Tuesday, February 28:
Thursday, March 2:
Tuesday, March 7:
Thursday, March 9:
Spring Break
Tuesday, March 21:
Thursday, March 23:
Tuesday, March 28:
Thursday, March 30:
Tuesday, April 4:
Thursday, April 6:
Tuesday, April 11:
Thursday, April 13:
Tuesday, April 18:
Thursday, April 20:
Tuesday, April 25:
Thursday, April 27:
Problem Sets, Exam
There will be a problem set due roughly every other week, and a
take-home final exam.
Problem
set 1: due Tuesday, January 31.
Problem set 2: due Tuesday, February 14.
Problem set 3: due Tuesday, February 28.
Problem set 4: due Tuesday, March 21.
Problem set 5: due Tuesday, April 4.
Problem set 6: due Tuesday, April 18.
References
Background:
Some notes about background for this course:
Lie
groups and Lie algebras
Representation
theory
Simple
quantum mechanical systems
Some sources for much of the material from the first semester
that we'll be using are:
John Morgan's web-page here has
lecture notes from the first semester.
Eckhard Meinrenken's lecture notes on Lie
Groups and Lie Algebras.
Alexander Kirillov, Jr.
An
Introduction to Lie Groups and Lie Algebras
Cambridge University Press, 2008
Note that the electronic version of this book is available freely
for Columbia students at the link above or via its entry in the
Columbia library catalog.
Much of the first seven chapters in this book were covered during
the first semester. Later on in the course we'll cover what is in
chapter 8.
Textbooks:
Some textbooks of various sorts that may be useful:
Woit, Quantum
theory, groups and representations
Carter, Segal and Macdonald, Lectures on Lie groups and Lie
algebras
Knapp, Lie Groups: Beyond an Introduction
Fulton and Harris, Representation theory
Bump, Lie groups
Sepanski, Compact Lie groups
Wallach, Symplectic geometry and Fourier analysis
Kirillov, Lectures on the orbit method
Online Lecture Notes:
The following selection of on-line lecture notes and course
materials may be useful:
Lecture notes from Andrei Okounkov's 2016-7 course: Fall
2016, Spring
2017
Berkeley
Lectures on Lie Groups and Quantum Groups
David Ben-Zvi course on representations of SL2, see notes
on this page.
Articles:
Representations
of Lie groups and the orbit method, Michele Vergne
Other sources:
Mikhail Khovanov's web page
of links.