## Math 274 - Quantum Groups

Fall, 2004

**Time and place:** MWF 3-4pm, 7 Evans Hall
**Course control number:** 55158

**Professor:** Mark Haiman

**Office:** 771 Evans

**Office hours:** Tues 1-3pm

**E-mail:**

**Phone:** (510) 642-4318

**Syllabus:**

- Review of classical reductive algebraic groups and Lie groups,
and their Lie algebras.
- Universal enveloping algebras.
- Kac-Moody algebras and their integrable representations.
- Quantum
*sl*_{2}.
- Quantized Kac-Moody algebras; Drinfeld's and Lusztig's formulations.
- Littlemann paths.
- Crystal bases (Kashiwara) and canonical bases (Lusztig).
- Geometric construction of canonical bases.
- Structure of (quantum) affine algebras and their representations.
- 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:

Homework will not be graded as such, but you may hand in any solutions
that you would like me to comment on (no promises on how long I will
take to do so).

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