Lie Groups and Representations: Mathematics G4343-4

Monday and Wednesday 11:00-12:15pm
Mathematics 507

This course will cover various aspects of the theory of Lie groups and their representations.
It is aimed at mathematics graduate students although graduate students in physics might
also find it of interest.

Tentative Syllabus

Problem Sets

Problem Set 1 (due Monday, September 24)
Problem Set 2 (due Monday, October 8)
Problem Set 3 (due Wednesday, October 24)
Problem Set 4 (due Monday, November 12)
Problem Set 5 (due Monday, December 3)
Problem Set 6 (due Monday, February 21)
Problem Set 7 (due Monday, March 24)
Problem Set 8 (due Monday, April 14)


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

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

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.

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

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.

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.

Old Lecture Notes

Some lecture notes from earlier versions of the course, when I was just teaching the
spring semester.  This year I'll be teaching the full year, so will cover a wider range
of material, including a more algebraic point of view, and more about non-compact groups.

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:

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.