Math 715 Winter 2015
Quantum groups and crystal bases

Lectures: TTh 1-2:30 in 3096 EH

Instructor: Thomas Lam, 2834 East Hall, tfylam@umich.edu

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

Course Homepage: http://www.math.lsa.umich.edu/~tfylam/Math715.html

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:

• Lecture 1 (Jan 8): Quantum Yang-Baxter relation and the HOMFLY polynomial
• Lecture 2 (Jan 13): Poisson algebras and deformations of commutative algebras
• Lecture 3 (Jan 15): Poisson manifolds and Poisson-Lie groups
• Lecture 4 (Jan 20): Lie bialgebras
• Lecture 5 (Jan 22): Manin triples
• Lecture 6 (Jan 27): Standard Lie bialgebra structure on simple Lie algebras
• Lecture 7 (Jan 29): Coboundary Lie bialgebras
• No Lecture on Feb 3
• Lecture 8 (Feb 5): Lie bialgebra structures on simple Lie algebras
• Lecture 9 (Feb 10): CYBE and integrable systems
• Lecture 10 (Feb 12): Hopf algebras
• Lecture 11 (Feb 17): Monoidal categories
• Lecture 12 (Feb 19): Coboundary and quasi-triangular Hopf algebras
• Lecture 13 (Feb 24): Deformations of Hopf algebras
• Lecture 14 (Feb 26): Quantizations of coPoisson Hopf algebras
• Lecture 15 (Mar 10): Quantized universal enveloping algebras
• Lecture 16 (Mar 12): Braid group action. Formula for R-matrix.
• Lecture 17 (Mar 17): Representations of U_q(sl_2).
• Lecture 18 (Mar 19): More representations of U_q(sl_2).
• Lecture 19 (Mar 24): Verma modules and finite-dimensional simples of U_q(g).
• Lecture 20 (Mar 26): Semisimplicity of Rep U_q(g). Crystal bases.
• Lecture 21 (Mar 31): Crystal basis of the lower half of U_q(g).
• Lecture 22 (Apr 2): Tensor products of crystal bases.
• Lecture 23 (Apr 7): Start Kashiwara's grand loop.
• Lecture 24 (Apr 9): Continue induction.
• Lecture 25 (Apr 14):
• Lecture 26 (Apr 16):
• Lecture 27 (Apr 21):