# From the drawers of the museum

One of my amateur interests is paleontology. Paleontologists looking for new examples have two options: go out in the field and dig up a new example, or go looking through drawers of museums to find old examples that had been overlooked. In this blog post I want to give an interesting example of the latter kind of research being useful in mathematics. Namely in discussions with Zhengwei Liu, we realized that an old example of Ocneanu’s gives an answer to a question that was thought to be open.

One of the central problems in fusion categories is to determine to what extent fusion categories can be classified in terms of finite groups and quantum groups (perhaps combined in strange ways) or whether there are exceptional fusion categories which cannot be so classified. My money is on the latter, and in particular I think extended Haagerup gives an exotic fusion category. However, there are a number of examples which seem to involve finite groups, but where we don’t know how to classify them in terms of group theoretic data. For example, the Haagerup fusion category has a 3-fold symmetry and may be built from $\mathbb{Z}/3\mathbb{Z}$ or $S_3$ (as suggested by Evans-Gannon). The simplest examples of these kind of “close to group” categories, are called “near-group categories” which have only one non-invertible object and have the fusion rules

$X^2 \cong X^{\oplus n} + \sum_g g$

for some group of invertible objects $g$. A result of Evans-Gannon (independently proved by Izumi in slightly more generality), says that outside of a reasonably well understood case (where $n = \#G -1$ and the category is described by group theoretic data), we have that $n$ must be a multiple of $\# G$. There are the Tambara-Yamagami categories where $n = 0$, and many examples (E6, examples of Izumi, many examples of Evans-Gannon) where $n = \#G$

Here’s the question: Are there examples where n is larger than $\# G$?

It turns out the answer is yes! In fact the answer is given by the $0$-graded part of the quantum subgroup $E_9$ of quantum $SU(3)$ from Ocneanu’s tables here. I’ll explain why below.

# When confusions annihilate

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

# Cyclotomic integers, fusion categories, and subfactors (March)

Frank Calegari, Scott Morrison, and I recently uploaded to the arxiv our paper Cyclotomic integers, fusion categories, and subfactors. In this paper we give two applications of cyclotomic number theory to quantum algebra.

1. A complete list of possible Frobenius-Perron dimensions in the interval (2, 76/33) for an object in a fusion category.
2. Given a family of graphs G_n obtained from a graph G by attaching a chain of n edges to a chosen vertex, an effective bound on the greatest n so that G_n can be the principal graph of a subfactor.

Neither of these results look like they involve number theory. The connection comes from a result of Etingof, Nikshych, and Ostrik which says that the dimension of every object in a fusion category is a cyclotomic integer.

A possible subtitle to this paper is

What’s so special about $(\sqrt{3} + \sqrt{7})/2$?

# New Journal: Quantum Topology

The European Math Society Publishing House (a non-profit publishing company which also publishes the Journal of the EMS, CMH, and half a dozen other journals) just announced a new journal: Quantum Topology. I think this is very exciting as it fills a nice hole in the existing journal options. The list of main topics include knot polynomials, TQFT, fusion categories, categorification, and subfactors. So there should be lots of material of interest to people here.

# Extended Haagerup Exists!

Following on from Noah’s post about the great Modular Categories conference last weekend in Bloomington, I’ll say a little about the talk I gave: Extended Haagerup exists!

The classification of low index, finite-depth subfactor planar algebras seems to be a difficult problem. Below index 4, there’s a wonderful ADE classification. The type A planar algebras are just Temperley-Lieb at various roots of unity (and so the same as $U_q(sl_2)$, as long as you change the pivotal structure). The type D planar algebras (with principal graphs the Dynkin diagrams $D_{2n}$) were the subject of Noah’s talk at the conference, and the $E_6$ and $E_8$ planar algebras are nicely described in Stephen Bigelow’s recent paper.

But what happens as we go above index 4? In 1994 Haagerup gave a partial classification up to index $3+\sqrt{3} \equiv 4.73205$. He showed that the only possible principal graphs come in two infinite families

and

(in both cases here the initial arm increases in steps of length 4) and another possibility

This result really opened a can of worms. Which of these graphs are actually realised? (Hint, they’re nicely colour-coded :-) What about higher index? What does it all mean? Are these graphs part of some quantum analogue of the classification of finite simple groups? Read one for the answer to the first question, at least.

# SF&PA: Subfactors = finite dimensional simple algebras

Since my next post on Scott’s talk concerns the construction of a new subfactor, I wanted to give another attempt at explaining what a subfactor is. In particular, a subfactor is just a finite-dimensional simple algebra over C!

Now, I know what you’re thinking, doesn’t Artin-Wedderburn say that finite dimensional algebras over C are just matrix algebras? Yes, but those are just the finite dimensional algebras in the category of vector spaces! What if you had some other C-linear tensor category and a finite dimensional simple algebra object in that category?

Let me start with an example (very closely related to Scott Carnahan’s pirate post).

# TQFTs via Planar Algebras (Part 3)

This is the third and final post in my series about using planar algebras to construct TQFTs. In the first post we looked at the 2D case and came up with a master strategy for constructing TQFTs. In the last post we began carrying out that strategy in the 3-dimensional setting, but ran into some difficulties. In this post we will overcome those difficulties and build a TQFT.

# TQFTs via Planar Algebras (Part 2)

In my last post I explained a strategy for using n-dimensional algebraic objects to construct (n+1)-dimensional TQFTs, and I went through the n=1 case: Showing how a semi-simple symmetric Frobenius algebra gives rise to a 2-dimensional TQFT. But then I had to disappear and go give my talk. I didn’t make it to the punchline, which is how planar algebras can give rise to 3D TQFTs!

In this post I will start explaining the 3D part of the talk. I won’t be able to finish before I run out of steam; that will have to wait for another post. But I will promise to use lots of pretty pictures!

# TQFTs via Planar Algebras

So today I am giving a talk in the Subfactor seminar here at Berkeley, and I thought it might by nice to write my pre-talk notes here on the blog, rather then on pieces of paper destined for the recycling bin.

This talk is about how you can use Planar algebras planar techniques to construct 3D topological quantum field theories (TQFTs) and is supposed to be introductory. We’ve discussed planar algebras on this blog here and here.

So the first order of buisness: What is a TQFT?

# Two fun problems

One of the points of this blog is for us to share the little problems we’d be discussing at tea if we were all still in Berkeley.  Here are two that came up in the last couple weeks.

As we all know, you can never know too much linear algebra.  So here’s a fun little linear algebra exercise that Dave Penneys asked us over beers on friday:  “Which matrices have square roots?”

The second question I don’t know the answer to, but I haven’t looked too hard.  The other week Penneys and I were trying to compute an example in subfactors and stumbled on the following interesting question about infinite groups (somewhat reminiscent of this old post). When can you find a group G and a proper inclusion G->G such that the image is finite index?

There’s the obvious example Z.  But once you start adding adjectives it starts getting tricky.  We were looking for a finitely generated group all of whose nontrivial conjugacy classes are infinite.  If only I knew more geometric group theory…