Coincidences of tensor categories March 22, 2009Posted by Noah Snyder in Category Theory, quantum groups, small examples, talks.
This week Scott and I were at a wonderful conference on Modular Categories at Indiana University. I find that I generally enjoy conferences on more specific subjects, especially in algebra. Otherwise you run the danger of every talk starting by defining some algebra you’ve never heard of (and won’t hear of again the rest of the conference) and then spend a while proving some properties of this random algebra that you still don’t know why you care about (let alone why you should learn about its projective modules). With more specific conferences if you don’t quite get something the first time you have a good change of seeing it again and it slowly sinking in. The organizers (Michael Larsen, Eric Rowell, and Zhenghan Wang) did an excellent job putting together and interesting, topically coherent, and fun conference. I was also pleasantly surprised by Bloomington, which turned out to actually be kind of cute. I have several posts I’d like to give on other people’s talks, in particular there were several talks (by Davydov, Mueger, and Ostrik) on the “Witt group” which involves the simplest case of a kind of cohomology of the periodic table of n-categories and thus should appeal to all of you over at the n-category theory cafe. But I think I’ll start out with our talks (which Scott and I prepared jointly based on our joint work with (Emily Peters)^2 and Stephen Bigelow).
The first of these talks (click for beamer slides) was on coincidences of small tensor categories. The strangest thing about this talk was that I was introduced as a “celebrity math blogger.”
Please note that in the slides I’ve completely elided the distinctions between a quantum group, its category of representations, and (when q is a root of unity) its semisimplified category of representations (where you quotient out by the kernel of the inner product as in David’s post).
The story that I wanted to tell in this talk is that there are coincidences in quantum group theory that look like the small coincidences of finite groups (like the isomorphism between PSL_2(F_5) and A_5, or the outer automorphism of S_6). Together with Emily Peters we described a bunch of coincidences involving the “even part of the subfactor planar algebra D_2n”. This is one of the very simplest braided tensor categories, it’s described very explicitly in another paper of ours. You should think of it as a “quantum subgroup of so(3)” much like the dihedral groups are “subgroups of so(3).”
We prove these coincidences in two ways. First we give a simple recipe (based on work of Kauffman, Kazhdan, Kuperberg, Wenzl and others) for recognizing when some category you’ve run across is going to appear as some sort of “quantum subgroup of quantum sl, so, sp, or g_2.” This gives a by-hands way of noticing several coincidences. Then we go back and re-explain most of them using some general techniques (mostly “level-rank duality” which is something that seems really quite mysterious until you read the papers of Beliakova and Blanchet and then you can distill it into two slides one of which is a pretty picture). Finally we illustrate these coincidences with beautiful pictures (thanks to Scott, google sketch-up, and adobe illustrator) and give applications to knot invariants.
Next up, our other talk: “Extended Haagerup exists!” (joint with Emily Peters and Stephen Bigelow).