Lie Groups and Representations: Mathematics G4344

Monday and Wednesday 1:10-2:25pm
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.  The standard syllabus for this course was set up as part of the
department's VIGRE program.  This semester I'll be emphasizing the more geometric
aspects of representation theory, as well as their relationship to quantum mechanics.

Tentative Syllabus

Lecture Notes

Cultural Background

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

Problem Sets

Problem Set 1 due Wednesday, February 12
Problem Set 2 due Monday, March 3
Problem Set 3 due Monday, March 31
Problem Set 4 due Wednesday, April 23

Final Exam


The following books are on reserve and 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.

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.

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

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

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

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

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