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.


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

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) 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.