Math 274 - Quantum Groups
Time and place: MWF 3-4pm, 7 Evans Hall
Course control number: 55158
Professor: Mark Haiman
Office: 771 Evans
Office hours: Tues 1-3pm
Phone: (510) 642-4318
- Review of classical reductive algebraic groups and Lie groups,
and their Lie algebras.
- Universal enveloping algebras.
- Kac-Moody algebras and their integrable representations.
- Quantum sl2.
- 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.
- 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
Uh(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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