Lie Groups and Representations:
Mathematics G4343 (fall 2013)
Monday and Wednesday 10:10-11:25am
507 Mathematics
This course will cover various aspects of the theory of Lie groups,
Lie algebras and their representations. It is aimed at
first-year mathematics graduate students although students in
physics might also find it of interest. This semester, the
emphasis will be on Lie algebras and the classification of
representations of finite-dimensional complex semi-simple Lie
algebras, covering most of the material in the Kirillov
textbook. This will be supplemented with various other topics,
including examples from quantum mechanics, sometimes extracted from
the notes on quantum mechanics and representation theory available here.
Problem Sets
There will be problem sets due roughly every other week, and a final
exam.
Problem
Set 1. Due Monday, September 23.
Problem
Set 2. Due Monday, October 7.
Problem
Set 3. Due Monday, October 21.
Problem
Set 4. Due Monday, November 11.
Problem
Set 5. Due Monday, November 25.
Problem
Set 6. Due Monday, December 9.
Textbook
Alexander Kirillov, Jr.
An Introduction to Lie Groups and
Lie Algebras
Cambridge University Press, 2008
Note that electronic version of this book is available freely for
Columbia students at the link above or via its entry in the Columbia
library catalog.
Another strongly recommended source is Eckhard Meinrenken's lecture
notes on Lie
Groups and Lie Algebras. In particular, for the early
part of the course, where our discussion and the one in Kirillov is
rather sketchy, these notes give details with a careful attention to
the confusing issues of left-versus-right actions and actions on
spaces versus actions on functions on the spaces.
Other References
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.
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.
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.
Lecture Notes
Background
on Classification of Lie Groups and Lie Algebras
Generalities
About Representation Theory
Induced
Representations and Frobenius Reciprocity
The
Peter-Weyl Theorem
Highest-Weight
theory: Verma modules
Highest-Weight
theory: Borel-Weil theorem
The
Weyl Character Formula
Adjoint Orbits and the Chevalley restriction theorem
The Harish-Chandra Isomorphism
Lie
algebra cohomology and the Borel-Weil-Bott theorem
Clifford
Algebras and Spin Groups
The
Spinor Representation
The Heisenberg Algebra and Metaplectic Group
The Metaplectic Representation
Geometric Quantization and the Orbit Method
The Dirac Operator and Representation Theory
Generalities about Representations of Real Semi-simple Lie Groups
SL(2,R)
SL(2,R) representations: Lie algebra methods
SL(2,R) representations: Parabolic induction and Discrete Series.
Old Lecture Notes
Some lecture notes from two earlier versions of the course.
I'm hoping to find time to revise some of these this spring, and
will also hope to write up notes for some topics we'll be covering
that were not in the earlier notes.
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:
Berkeley
Lectures on Lie Groups and Quantum Groups
Representation
Theory
Course by Constantin Teleman
David Ben-Zvi course on representations of SL2. Part
1, Part
2, Part
3.
Eckhard Meinrenken lecture notes on Lie
Groups and Lie Algebras.