Columbia University G4343
Lie Groups and Representations

Basic information

Call number:	47746
Room/Time:	MW 11am--12:15pm, 307 Math
Instructor:	Mikhail Khovanov
Office:	517 Math
Office Hours:	Walk-in or by appointment
E-mail:	khovanov@math.columbia.edu

Discussions:	Thursday 2:30-3:30pm, room 528
TA:	Alexander Ellis, ellis@math.columbia.edu
Webpage:	www.math.columbia.edu/~khovanov/lgroups

Syllabus

Representation theory of finite groups.
McKay correspondence.
Representation theory of the symmetric group.
Basics of Lie groups and Lie algebras.
Solvable and nilpotent Lie algebras.
Universal enveloping algebras. PBW theorem.
sl(2) and its representations.
Classification of simple Lie algebras. Dynkin diagrams.
Representations of sl(n) and Schur-Weyl duality.
Representations of simple Lie algebras. Complete reducibility.
Haar measure. Representations of compact Lie groups.

Online resources:

M. Alexandrino and R. Bettiol, Introduction to Lie groups, adjoint action and its generalizations
J. Gallier, Notes on Lie group actions: manifolds, Lie groups and Lie algebras
A. Kirillov, Jr., Introduction to Lie groups and Lie algebras
S. Sternberg, Lie Algebras
D. Milicic, Lectures on Lie Groups
Notes for Lie algebras class by Victor Kac.

Brian Hall, An Elementary Introduction to Groups and Representations
Peter Woit, Lie groups and representations
Hans Samelson, Notes on Lie algebras
Eckhard Meinrenken, Clifford algebras and Lie groups
A brief summary Root systems and Weyl groups, by Jeffrey Adams.
Online notes for MIT course Introduction to Lie groups
Vera Serganova, Representation theory: representations of finite groups, symmetric groups, GL(n), quivers.

Representation theory overview:

Constantin Teleman, Representation theory
P.Etingof et al., Introduction to representation theory.

Books:

The following books will be placed on reserve in the math library:
J.E.Humphreys, Introduction to Lie algebras and representation theory.
W.Fulton and J.Harris, Representation Theory: A First Course.
R. Carter, G. Segal, and I. MacDonald, Lectures on Lie Groups and Lie Algebras. A good supplementary reading for our course is Chapter II, by Segal.

Additional books on Lie groups and Lie algebras:
Daniel Bump, Lie groups, Graduate Texts in Mathematics, Vol. 225.
T.Brocker and T.Dieck, Representations of Compact Lie Groups.
Anthony Knapp, Lie groups, Lie algebras, and cohomology.
Anthony Knapp, Lie groups Beyond an Introduction.
J.-P. Serre, Complex Semisimple Lie algebras.