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

Textbooks

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.