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)

Textbooks

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

representations.

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.