Mini-course on Quantum Groups
This is an ongoing series of 12 lectures on quantum groups I am giving at the Perimeter Institute (as part of the 'Grad Talks' series). It started by a couple of PI grad students asking me to teach them about quantum groups in a casual conversation, and it turned into a long-ish open-to-all lecture series. The talks take place on (most) Mondays at 11 a.m. in the Alice Room (usually).
The lectures are intended as an introduction to quantum groups, building up to some important developments of the theory. I hope that these lectures will be somewhat helpful for students working in quantum information and quantum gravity, and will be interesting to others. And I hope that my presentation and organization skills will improve as the series grow. Below I briefly summarize each lecture, with lecture notes attached as they become ready. Thanks to Jacob Stauttener and PI, video recordings of the lectures are available online (on PIRSA). Please don't hesitate to contact me (see below for contact info) if you have questions or suggestions about these lectures.
Speaker: Lucy Liuxuan Zhang (lzhang *at^ math *dot^ utoronto *dot^ ca).
Content of past talks
- Lecture 1 (February 2, 2009): Hopf algebras with a pair of examples, algebra/coalgebra duality.
- Lecture 2 (February 9, 2009): important examples of Hopf algebras -- the universal enveloping algebra of a Lie algebra, and the set of regular functions on an algebraic group.
- PIRSA recording #09020034
(Note: In this recording, I messed up the part on algebraic groups and regular functions on them pretty badly. For a cleaner presentation, please see recording for the next lecture. Though the notes here are fine.)
- Lecture 3 (February 23, 2009): Hopf algebra structures on the universal enveloping algebra of sl(2), and on the set of regular functions on SL(2); modules and comodules
- Lecture 4 (March 9, 2009): module/comodule duality; the big picture -- categorical language as a bridge between algebra and topology; category basics -- categories, functors, natural transformations, tensor categories.
- Lecture 5 (March 16, 2009): Braided bialgebras, braided categories, building up to the theorem connecting these two.
- Lecture 6 (March 23, 2009): Proof of the theorem connecting braided bialgebras and braided categories; all the topological objects we shall need -- tangles, braides, framed tangles, and their isotopy classes; categorical definition of isotopy classes of tangles -- the Tangle Category.
- Lecture 7 (March 30, 2009): Categorical definition of isotopy classes of braids -- the Braid Category, connection between braided categories and isotopy invariants of braids; braided categories with duals, connection between Hopf algebras and tensor categories with dualities.
- Lecture 8 (April 6, 2009): Ribbon algebras, ribbon categories, connection between these and isotopy invariants of ribbons or framed tangles (including framed links); quantum trace and quantum dimension.
- Lecture 9 (April 13, 2009):
Plan for future talks
- Lecture 10-12: deformation theory, quantum universal enveloping algebras, existence and uniqueness of quantization of a semisimple Lie algebra. Maybe also the KZ equations and the Drinfeld-Kohno Theorem, but no promises.