# 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.

# Knot polynomial identities and quantum group coincidences (February)

Scott Morrison, Emily Peters and I have just uploaded to the arxiv our paper Knot polynomial identities and quantum group coincidences. In this paper we prove several new strange identities between certain specializations of colored Jones polynomials and other classical knot polynomials. For example, we prove that for any knot (but not for links!) the 6th colored Jones polynomial at a 28th root of unity is twice the value of a certain specialization of the HOMFLYPT polynomial (for the exact formulas see the first page of the paper).

Each of these identities of knot polynomials comes from a coincidence of small tensor categories involving the even part of one of the $D_{2n}$ subfactors. For example, the above identity comes from an equivalence

$\frac{1}{2} \mathcal{D}_{8} \cong \text{Rep}^{uni}{U_{s=\exp({2 \pi i \frac{5}{14} })}(\mathfrak{sl}_4)}^{modularize}$

To recover the knot polynomial identity, one computes the Reshetikhin-Turaev invariant for a particular object on the left (getting half the relevant colored Jones polynomial for knots, but something worse for links) and for the corresponding object on the right (getting the specialization of HOMFLYPT).

(In that equation above there’s a lot of technical terms on the right side. “Uni” means we’re using Turaev’s unimodal pivotal structure instead of the usual pivotal structure, and “modularize” means take the Bruguieres-Mueger modularization where you add isomorphisms between the trivial object and all simple objects that “behave like the trivial object.”)

For most of these coincidences we give no fewer than three separate proofs, as well as an exciting diagram which explains the proof in pictures. After the jump I’ll sketch the flavor of these different arguments.

One of the biggest difficulties in writing this paper was getting all our conventions straight because we constantly need to jump between different quantum groups, and knot polynomials. Life becomes very messy if you’re using one convention for q in one place and a different one somewhere else. So one goal of this paper is to be “full of correctness.” We hope that if you ever wanted to know about the relationship between quantum groups and diagramatics you can just look at this paper and have nice consistent conventions for everything. In particular, if you notice even small errors we’d really love to hear about them so that they can be fixed.

# A hunka hunka burnin’ knot homology

One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?

This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.

At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong). Continue reading

# The Witt group, or the cohomology of the periodic table of n-categories

A very popular topic at the Modular Categories conference was the a generalization of the Witt group which is being developed by Davydov, Mueger, Nikshych, and Ostrik. What is this Witt group? Well it’s the simplest case of the cohomology of the periodic table of n-categories!

In this post I want to explain the definition of this cohomology theory and explain why it generalizes the classical Witt group.

First recall the Baez-Dolan periodic table.

Periodic Table

# Hall algebras and Donaldson-Thomas invariants I

I would like to tell you about recent work of Dominic Joyce and others (Bridgeland, Kontsevich-Soibelman, Behrend, Pandaripande-Thomas, etc) on Hall algebras and Donaldson-Thomas invariants.  I don’t completely understand this work, but it seems very exciting to me. This post will largely be based on talks by Bridgeland and Joyce that I heard last month at MSRI.

In this post, I will concentrate on different versions of Hall algebras. Let us start with the most elementary one. Suppose I have an abelian category $\mathcal{A}$ which has the following strong finiteness properties: namely $Hom(A,B)$ and $Ext^1(A,B)$ are finite for any objects $A, B$. Then one can define an algebra, called the Hall algebra of $\mathcal{A}$, which has a basis given by isomorphism classes of objects of $\mathcal{A}$ and whose structure constants $c_{[M], [N]}^{[P]}$ are the number of subobjects of $P$ which are isomorphic to $N$ and whose quotient is isomorphic to $M$.

The main source of interest of Hall algebras for me is the Ringel-Green theorem which states that if you start with a quiver $Q$, then the Hall algebra of the category of representation of $Q$ over a finite field $\mathbb{F}_q$ is isomorphic to the upper half of the quantum group corresponding to $Q$ at the parameter $q^{1/2}$.

The obvious question concerning Hall algebras is to come up with a framework for understanding them when the Hom and Ext sets are not finite. This is what Joyce has done and he has applied it where $A$ is the category of coherent sheaves on a Calabi-Yau 3-fold.

# Coincidences of tensor categories

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).

# Group rings arrr commutative

If you are familiar with group rings, you might think that the title of this post is false. If G is a nonabelian group, multiplying the basis elements g and h in $\mathbb{Z}G$ can yield $gh \neq hg$, so we have a problem. In general, if you have a problem that you can’t solve, you should cheat and change it to a solvable one (According to my advisor, this strategy is due to Alexander the Great). Today, we will change the definition of commutative to make things work.

# Representations of the small quantum group

Dear Internets,

I’m sorry for having disappeared for so long, and I promise I’ll get back to my series on planar algebras soon. I’ve been busy writing papers and couldn’t really justify to my coauthors why I’m not writing our papers and instead writing math on the internet. However, Saturday I was in San Francisco without access to LaTeX, and Malia was at her spinning guild, so here’s a post!

Yours,

Noah

P.S. Below the break I’ll walk you through understanding much of the representation theory of the small quantum group for $\mathfrak{sl}_2$ at a third root of unity, assuming you already understand the representation theory of the usual lie algebra $\mathfrak{sl}_2$. It’s an interesting example to work through because the representation theory is not semisimple. There will be lots of fun pictures. The choice of a third root of unity isn’t important here, everything would work similarly for any odd root of unity.

# Quantum Topology in Hanoi

I recently got back from an interesting trip to Vietnam, where I attended “Quantum Topology in Hanoi“. This conference was held at “VAST”, the Vietnamese Academy of Science and Technology, and organised by Thang Le and Stavros Garoufalidis of Georgia Tech.

Being in Vietnam was great fun, and most participants (including at least one of our readers) enjoyed the crazy bustle of life in Hanoi. Even the afternoon when the power went out, the backup generator failed, the airconditioning was off, and Dylan Thurston talked about the combinatorial model for knot Floer homology in 34°C wasn’t so bad. :-) After the conference, I went to Halong Bay with Dylan and Jana for the weekend, and then up to Sapa for two more days.

My talk was about the “lasagna operad” acting on Khovanov homology, and I decided to try Dror’s ‘single page handout’ approach, and even more adventurously, his freshly made ‘javascript handout browser’. You can see my slide, and Dror’s programming, here.

But what I’d like to talk about now is Nathan Geer’s talk, about “fake quantum dimensions”. I’ll start somewhere near the beginning, reminding you how to build tangle invariants out of braided tensor categories, then explain what goes wrong when quantum dimensions become zero, and finally what Nathan and co. propose to do about it. Continue reading

# Questions about the MOY relations

I have some rather specific questions about the MOY knot invariant which have come up as Sabin Cautis and I have been thinking about Khovanov-Rozansky homology. I’ll start by explaining the following “theorem” and then I’ll ask some questions about it. Hopefully someone (for eg. Scott) will be able to answer them.

Consider closed crossingless Reshetikhin-Turaev diagrams for sl(n) where all strands are labelled by 1 or 2. This means that we look at oriented planar graphs with all edges either labelled 1 or 2, and each vertex trivalent with either two single edges and one double edge and matching orientations.

We can think of such a graph as being related to the representation theory of (quantum) sl(n). The single strands correspond to the standard representation $\mathbb{C}^n$ and the double strands to $\Lambda^2 \mathbb{C}^n$ The vertices correspond to morphisms of representations (for example $\mathbb{C}^n \otimes \mathbb{C}^n \rightarrow \Lambda^2 \mathbb{C}^n$). Within this representation theory context, each graph G can be evaluated to a Laurent polynomial c(G) (an element of the trivial representation of quantum sl(n)).