MATH 8300 - Topics in Algebra: Quantum Groups - Spring 2020

| Class Lecture Notes | Homework | Reading |

General Information Time MWF, 1:25 p.m. -- 2:15 p.m. Location Vincent Hall, Room 2 Professor Ben Brubaker (brubaker@math.umn.edu) Office: Vincent Hall, Room 352 Office Phone: 612-625-7396 Office Hours: MWF 11:00-12:00, and always available by appointment Textbook None required, we use a combination of notes prepared by me and small segments of several textbooks (see "Further Reading" below) Course Assignments 3 optional problem sets over the semester, an optional presentation on a special topic at the end of the course, no exams (Note: undergraduates enrolled will be required to submit problem sets and do a final presentation) Course Ad Linked here as a pdf. Homework There will be 3 problem sets over the course of the semester (roughly one every 4 weeks) posted here. Problem Set 1, due Friday, February 21. (TeX source file) Problem Set 2, due Friday, March 27. (TeX source file) Reading Materials As noted above, there is no required textbook. I will post scans of my own course notes on the Class Lecture Notes page, linked above. For the most part, these will be inspired by the following sources: "Quantum Groups" by Christian Kassel (Springer GTM 155) "A Quantum Groups Primer" by Shahn Majid (London Math. Soc. Lecture Note Series 292) - this book has real insight and is nicely divided into individual lectures of reasonable size. Better still, UMN students have online access to the book through the library website. "Lectures on Quantum Groups" by Jens Carsten Jantzen (AMS Graduate Studies, Vol. 6) "A Guide to Quantum Groups" by V. Chari and A. Pressley (Cambridge Univ. Press) Many of the above sources assume a familiarity with representation theory and algebraic groups that we won't assume! In particular, throughout the course, we will remind the audience about basic notions in representation theory of algebras. See for example: Peter Webb's book on Representation Theory is one of the sources we'll use to discuss representation theory of groups and group algebras. Drozd and Kirichenko's "Finite Dimensional Algebras" has a nice discussion of many of the properties of semisimple algebras used at the outset of the class. (Since this linked pdf is from Drozd's homepage, I trust it is ok to distribute freely.) Online notes on quantum groups: Notes on Quantum Groups from a course given by my longtime collaborator Daniel Bump, at Stanford University in 2019. They are written as Beamer slides, and include a number of very nice pictures. Resources on evaluations in the 6-vertex model: Six-vertex, Loop, and Tiling Models: Integrability and Combinatorics, by Paul Zinn-Justin contains lots of pretty pictures to help understand solvable lattice models. In particular, we explained a proof of the ASM conjecture along the lines of pp. 24--27 in this paper. Schur Polynomials and the Yang-Baxter Equation, a paper by Brubaker, Bump, and Friedberg. It evaluates a general class of 6-vertex partition functions. The published version (in Comm. Math Phys.) does an even more general set of Boltzmann weights.