As mathematicians we spend most of our lives confused about something or other. Of course, this is occasionally interrupted by moments of clarity that make it worth it. I wanted to discuss a particularly pleasant circumstance: when two confusions annihilate each other. I’ll give two examples of times that this happened to me, but people are encouraged to provide similar examples in the comments.
In both cases what happened was that I had:
- A question to which I didn’t know the answer
- An answer to which I didn’t know the question
When talking about planar algebras one often runs into the unfortunate fact that sometimes you want to think about shaded planar algebras (which correspond to 2-categories with two objects) and sometimes you want to think about unshaded planar algebras (which correspond to monoidal categories, or 2-categories with one object). A natural question is given a shaded planar algebra can you turn it into an unshaded one, and if so in how many ways? I ran across this question after we wrote this paper (with Emily and Scott) where there’s two different unshaded D_2n planar algebras but only one shaded one (basically because -1 has two different square roots). A very natural question I really wanted to know the answer to was “what happens if you look at one of the simplest examples: group subfactors.” On the other hand, for a long time I’d wondered what a Tambara-Yamagami category was (since they come up as examples all over the case in the fusion category literature).
I went to a conference in Waco (run by Eric Rowell and Deepak Naidu). Friend of the blog David Jordan was speaking about his joint work with (former mathcamper!) Eric Larson, and to make a long story short, during the talk it became clear to me that TY categories are exactly the ones that come from unshaded versions of group subfactors.
The other example concerned GHJ (Goodman, de la Harpe, Jones) subfactors (which I didn’t understand the construction of), and the question (which I didn’t know the answer to) was “what subfactors do you get from the ADE-type module categories over quantum su(2)?” The tricky thing here is that the answer to the latter question is *not* the ADE subfactors! Instead in discussions after a talk of Wenzl‘s at the 2010 Shanks workshop at Vanderbilt, it became clear to me that again these two confusions anihilated. This story is a little harder to tell in a blog post, but Dave Penneys wrote up some notes of a talk I gave about this on a retreat.