Math 715 Winter 2015
Quantum groups and crystal bases

Lectures: TTh 1-2:30 in 3096 EH

Instructor: Thomas Lam, 2834 East Hall,

Office Hours: By appointment, or approach me after class.

Course Homepage:

Prerequisites: Familiarity with the structure theory and representation theory of complex semisimple Lie algebras.

Grading: There will be no compulsory homework. Optional exercises and problems will be given during lectures.

Exams: There will be no exams.

Synopsis: This course will be an introduction to the theory of quantum groups, canonical bases, and crystal graphs. Quantum groups, or quantized enveloping algebras, are Hopf algebras that deform the universl enveloping algebra of a complex semisimple Lie algebra. In the first half of the course I will discuss the definition and representation theory of quantum groups. In the second half of the course, I will discuss the "canonical bases" or "global bases". These are distinguished bases for irreducible representations of quantum groups, or for the upper-half of the quantized enveloping algebra. At q = 0, we obtain crystal graphs, which are a combinatorial model for irreducible representations of complex semisimple Lie algebras.

Textbook: For the first half of the course, my plan is to follow "A guide to Quantum Groups", by Vyjayanthi Chari and Andrew Pressley.
For the second half of the course, I hope to roughly follow "On crystal bases of the q-analogue of universal enveloping algebras" by Masaki Kashiwara (Duke Math. J. 1991).

Other reference books:
Brown and Goodearl, Lectures on Algebraic Quantum Groups
Etingof and Schiffmann, Lectures on Quantum Groups
Jantzen, Lectures on Quantum Groups
Joseph, Quantum Groups and Their Primitive Ideals
Kassel, Quantum Groups
Lusztig, Introduction to Quantum Groups

Basic material on Lie algebras:
Humphreys, Introduction to Lie Algebras and Representation Theory
Kac, Infinite dimensional Lie algebras

List of lectures: