Lie Groups and Representations: Mathematics G4344 (spring 2012)

Monday and Wednesday 1:10-2:25pm
Mathematics 507

This course will cover various aspects of the theory of Lie groups and their representations, following on from Andrei Okounkov's fall semester course.  It is aimed at mathematics graduate students although graduate students in physics might also find it of interest.

Teaching assistant is Alex Ellis, who will run a weekly discussion section.

Tentative Syllabus

Problem Sets

There will be problem sets due roughly every other week, and a final exam.

Problem Set 1
Due Monday February 6
Problem Set 2
Due Monday February 27
Problem Set 3
Due Wednesday March 21
Problem Set 4
Due Wednesday April 18


I won't be following closely any particular textbook, but during parts of the course I
will be using:

Knapp, Anthony W., Lie Groups: Beyond an Introduction (Second Edition)
Birkhauser, 2002.
The first half of this book contains a very careful discussion of many of the topics we
will be covering.

Carter, Roger, Segal, Graeme, and MacDonald, Ian,
Lectures on Lie Groups and Lie Algebras,
Cambridge University Press, 1995.
This book is at the other extreme from the book by Knapp, providing a quick sketch
of the subject.

Sepanski, Mark,
Compact Lie Groups,
Springer-Verlag, 2006.
This book gives a detailed discussion of one of our main topics, the representations of
compact Lie groups, leading up to the Borel-Weil geometrical construction of these

Alexander Kirillov, Jr.
An Introduction to Lie Groups and Lie Algebras
Cambridge University Press, 2008

The following books cover much of the material of this course, at more or less
the same level.

Rossman, Wulf,
Lie Groups,
Oxford University Press, 2002.

Fulton, William, and Harris, Joe,
Representation Theory: A First Course,
Springer-Verlag, 1991.

Hall, Brian,
Lie Groups, Lie Algebras, and Representations:  An Elementary Introduction
Springer-Verlag, 2003.

Bump, Daniel,
Lie Groups,
Springer 2004.
New 2011 edition available on-line.

Gurarie, David,
Symmetries and Laplacians
Dover paperback, 2008.

Kirillov, A. A.,
Lectures on the Orbit Method
AMS, 2004.

Brocker, Theodor and tom Dieck, Tammo,
Representations of Compact Lie Groups,
Springer-Verlag, 1985.

Adams, J. Frank,
Lectures on Lie Groups,
University of Chicago Press, 1969.

Goodman, Roe and Wallach, Nolan,
Representations and Invariants of the Classical Groups,
Cambridge University Press, 1998.

Lecture Notes

Background on Classification of Lie Groups and Lie Algebras
Generalities About Representation Theory
Induced Representations and Frobenius Reciprocity
The Peter-Weyl Theorem
Highest-Weight theory: Verma modules
Highest-Weight theory: Borel-Weil theorem
The Weyl Character Formula
Adjoint Orbits and the Chevalley restriction theorem
The Harish-Chandra Isomorphism
Lie algebra cohomology and the Borel-Weil-Bott theorem
Clifford Algebras and Spin Groups
The Spinor Representation
The Heisenberg Algebra and Metaplectic Group
The Metaplectic Representation
Geometric Quantization and the Orbit Method
The Dirac Operator and Representation Theory
Generalities about Representations of Real Semi-simple Lie Groups
SL(2,R) representations: Lie algebra methods
SL(2,R) representations: Parabolic induction and Discrete Series.

Old Lecture Notes

Some lecture notes from two earlier versions of the course.  I'm hoping to find time to revise some of these this spring, and will also hope to write up notes for some topics we'll be covering that were not in the earlier notes.

Cultural Background
Representations of Finite Groups: Generalities, Character Theory, the Regular Representation
Fourier Analysis and the Peter-Weyl Theorem

Lie Groups, Lie Algebras and the Exponential Map
The Adjoint Representation
More About the Exponential Map
Maximal Tori and the Weyl Group
Roots and Weights
Roots and Complex Structures
SU(n), Weyl Chambers and the Diagram

Weyl Reflections and the Classification of Root Systems
SU(2) Representations and Their Applications
Fundamental Representations and Highest Weight Theory

The Weyl Integral and Character Formulas
Homogeneous Vector Bundles and Induced Representations
Decomposition of the Induced Representation
Borel Subgroups and Flag Manifolds
The Borel-Weil Theorem
Clifford Algebras
Spin Groups
The Spinor Representation
The Heisenberg Algebra
The Metaplectic Representation
Hamiltonian Mechanics and Symplectic Geometry
The Moment Map and the Orbit Method
Schur-Weyl Duality
Affine Lie Algebras
Other Topics

Online Resources

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

Berkeley Lectures on Lie Groups and Quantum Groups

Representation Theory Course by Constantin Teleman

Dan Freed course on Loop Groups and Algebraic Topology

David Ben-Zvi course on representations of SL2.  Part 1, Part 2, Part 3.

Eckhard Meinrenken lecture notes on Lie Groups and Lie Algebras.

Eckhard Meinrenken lecture notes on Lie Groups and Clifford Algebras.