Lie Groups and Representations: Mathematics G4344

Monday and Wednesday 11:00-12:15pm
Mathematics 307

This course will cover various aspects of the representation theory of Lie groups.  It
is aimed at mathematics graduate students although graduate students in physics might
also find it of interest.  I'll be emphasizing the more geometric aspects of representation
theory, as well as their relationship to quantum mechanics.

The first semester of this course was taught by Prof. Mu-Tao Wang, and covered most
of the book Lie Groups, Lie Algebras and Representations, by Brian Hall (except for
sections 7.4-7.6).  I'll be assuming most of this material, with some review as needed.

Tentative Syllabus

Lecture Notes

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

Problem Sets

Problem Set 1  (due Monday, February 5)
Problem Set 2  (due Monday, February 19)
Problem Set 3  (due Monday, March 19)
Problem Set 4  (due Monday, April 16)

Old Lecture Notes

The following lecture notes were used when I last taught this course, in the spring of
2003.  This semester I'll be covering some of the same topics (as well as different ones)
so may use some of this material.  I also hope to write up some newer versions of these
notes.

Lie Groups, Lie Algebras and 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

Problem Sets

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

Textbooks

The closest thing to a textbook for the course will be the lecture notes by Graeme Segal
in:

Carter, Roger, Segal, Graeme, and MacDonald, Ian,
Lectures on Lie Groups and Lie Algebras,
Cambridge University Press, 1995.

For some of the topics to be covered in the first half or so of this semester, a good
detailed textbook is a very new one that just appeared:

Sepanski, Mark,
Compact Lie Groups,
Springer-Verlag, 2006.

I'll also be covering in much more detail the topics that are sketched in sections 7.4-7.6 of
the textbook used last semester:

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

The following books cover much of the material of this course, at more or less
the same level.  The first four are closest in spirit to what we will be covering.

Simon, Barry,
Representations of Finite and Compact Lie Groups,
AMS, 1996.

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

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

Bump, Daniel,
Lie Groups,
Springer 2004.

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

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

Lectures on Lie Groups,
University of Chicago Press, 1969.

Taylor, Michael,
Noncommutative Harmonic Analysis,
AMS, 1986.

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

Online Resources

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

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 Clifford Algebras.