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).
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. Notes
Thursday, February 2: Analogy with spin groups and the spinor
representation. Notes
Tuesday, February 7: Howe duality and the theta correspondence
Thursday, February 9: Heisenberg and metaplectic groups over
other fields
Tuesday, February 14: Theta functions and automorphic forms
Thursday, February 16: Review of classification theory of Lie
algebras, the universal enveloping algebra
Tuesday, February 21: Classification theory of semi-simple
complex Lie algebras
Thursday, February 23: Verma modules and highest-weight
representations
Tuesday, February 28: Class canceled due to illness
Thursday, March 2: More on Verma modules and highest-weight
representations
Tuesday, March 7: Infinitesimal character, Harish-Chandra
homomorphism
Thursday, March 9: Lie algebra cohomology: definition
Spring Break
Tuesday, March 21: Lie algebra cohomology: cohomology of
compact groups, Kostant's theorem
Thursday, March 23: Lie algebra cohomology: the Weyl
character formula
Tuesday, March 28: The orbit method and geometric quantization
Thursday, March 30: Geometric representation theory.
Borel-Weil theorem
Tuesday, April 4: Borel-Weil-Bott theorem
Thursday, April 6: Real semi-simple Lie groups and Lie algebras:
classification
Tuesday, April 11:
Thursday, April 13: SL(2,R) and its representations
Tuesday, April 18:
Thursday, April 20:
Tuesday, April 25: Relation to modular forms
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, March 7.
Problem
set 4: due Tuesday, March 28.
Problem set 5: due Tuesday, April 11.
Problem set 6: due Tuesday, April 25.
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.