Columbia University GR6344
Lie Groups and Representations II

Basic information

Call number: 62723
Room/Time: MW 2:40pm--3:55pm, 407 Math
Instructor: Mikhail Khovanov
Office: 620 Math
Office Hours: Walk-in or by appointment
TA: Yakov Kononov,

Syllabus (tentative)

McKay correspondence between finite subgroups of SU(2) and affine Dynkin diagrams.
Verma modules and classification of irreducible representations of simple Lie algebras. Casimir element and complete reducibility of representations. Center of the universal enveloping algebra and the Harish-Chandra theorem.
Finite-dimensional representations of simple Lie algebras. Kostant partition function. Weyl character formula.
Characters of irreducible representations of sl(n) and Schur functions. Schur-Weyl duality between representations of sl(n) and the symmetric group. Classification of symmetric group representations. Combinatorial formulae. Jucys-Murphy elements and Young basis in irreducible representations. Symmetric functions.
Structure and decomposition of tensor products of irreducible representations. Examples for sl(2) and for rank 2 Lie algebras.
Gelfand pairs.
Clifford algebras and spin representations.
Representation theory in the non-semisimple case. Representations of Artinian algebras. Projective functors in categories of highest weight representations.
If time allows: Hopf algebras and quantum groups. Coxeter groups and Hecke algebras. Simple Lie algebras over integers. Lie algebra cohomology.


J.Humphreys, Introduction to Lie algebras and representation theory.
A.Knapp, Lie groups: Beyond an introduction.
D.Bump, Lie groups.
W.Fulton and J.Harris, Representation Theory: A First Course.

Shlomo Sternberg, Lie Algebras
Vera Serganova, Representation theory: representations of finite groups, symmetric groups, GL(n), quivers.


Homework 1      

Homework 2      

Homework 3