Quantum Groups & Knot Invariants [1 of n]
This is the first installment of what I hope will eventually be a series of posts. I'll be making them over the next few months as I try to improve my understanding of the relationship between knot invariants and the representation theory of quantum groups. In particular, my minimal goal for the year is to learn how to calculate the Jones polynomial of a link via the representation theory of the q-deformed analogue of the universal enveloping algebra of sl_2 (where sl_2 is the Lie algebra of 2 by 2 matrices with zero trace).
While I hope these posts will be of interest to others (and I will try to write them with some sort of general reader in mind), I should admit up front that I'm mostly writing them for myself. I know (or used to know) a bit about both knot invariants and quantum groups separately, but it's been a while since I looked at either subject and I never really knew much about how they were connected. I'm challenging myself to try to improve on that state of affairs: first by refreshing my old knowledge and then by moving on to something new [or at least, new to me: I'm not doing anything original here].
My idea is that I'm more likely to achieve this goal if I have at least a notional audience to keep me honest about my progress. Hence, this blog and this series of posts.
(I'm also using this as an excuse to try to teach myself to make better use of LaTeX packages like TikZ and xypic. Since Tumblr really doesn't have any good way of rendering mathematics, you should expect to see a lot of embedded images in the posts in this series.)
The next few posts in this series will be my attempts to summarize the key ideas at work as best as I can. These are mostly ideas I'm already fairly familiar with (or at least, ideas I should be familiar with), so I hope this stage won't take more than a few weeks. The limiting factor might just be how much time I have to write things up properly.
Knots and links; their diagrams and the idea of knot invariants
The Jones polynomial via Kauffman's bracket polynomial
Braids, Hecke algebras of type A and the Temperley-Lieb algebra
[+ the Jones polynomial via a trace on the Temperley-Lieb algebra]
[+ monoidal categories; examples and basic properties]
Braided monoidal categories
Motivating bialgebras and Hopf algebras
Representations of some particular quantum groups
Universal R-matrices and quasitriangular Hopf algebras
Schur-Weyl duality (both classical and quantum)
I won't necessarily be posting about these topics in this order, but this order is at least as likely as any other. [Edit: any additions to the list will be indicated in square brackets like this, and prefaced with a + symbol.]
I'll update the list above as I finish writing the posts (both to add links and perhaps to slightly revise the list itself with new or different topics).
After that, what I post about next will depend on where my reading takes me and how much progress I manage to make. At a minimum, I'll try to post a quick summary of whatever I'm currently reading every couple of weeks.















