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
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
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.
Adams, J. Frank,
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.