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)
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.