Course control number: 55158

Professor: Mark Haiman
Office: 771 Evans
Office hours: Tues 1-3pm
E-mail:
Phone: (510) 642-4318

Syllabus:

1. Review of classical reductive algebraic groups and Lie groups, and their Lie algebras.
2. Universal enveloping algebras.
3. Kac-Moody algebras and their integrable representations.
4. Quantum sl₂.
5. Quantized Kac-Moody algebras; Drinfeld's and Lusztig's formulations.
6. Littlemann paths.
7. Crystal bases (Kashiwara) and canonical bases (Lusztig).
8. Geometric construction of canonical bases.
9. Structure of (quantum) affine algebras and their representations.
10. Conjectures and open problems.

Prerequisites: Good general algebra background.

Recommended reading:

• V. G. Drinfeld, Quantum groups, Proceedings of the ICM, Berkeley, 1986. Clear and concise survey of the philosophy and origins of the subject. Contains Drinfeld's original definition of U_h(g), also discovered independently by Jimbo.
• R. Carter, Finite Groups of Lie Type (Wiley, 1985; paperback 1993). Chapter 1 is an excellent short course on reductive algebraic groups.
• J. Humphreys, Linear Algebraic Groups (Springer 1975, 1981). More leisurely, extended treatment of algebraic groups. Also discusses Hopf algebras briefly.
• V. Kac, Infinite Dimensional Lie Algebras, 3rd Ed. (Cambridge Univ. Press, 1990). The standard reference for Kac-Moody algebras.
• J. C. Jantzen, Lectures on Quantum Groups (AMS Graduate Studies in Math., vol. 6, 1996). The best textbook for most of the material that we will cover in this course.
• G. Lusztig, Introduction to Quantum Groups (Birkhauser, 1993). The essential reference for Lusztig's geometric construction of canonical bases. Technical in style, omits motivation and examples.
• M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63, no. 2, 1991, pp 465-516. Definition and construction of crystal bases. Kashiwara's original paper is still the best reference.

Homework: Here are some exercises to try:

• 1-7
• 8-15