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 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.

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.

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.

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.

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.