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
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.