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
In quantum algebra we’re often studying some classical algebraic notion, but instead of working in the category of vector spaces you instead work in a more general tensor category. For example, the theory of finite type knot invariants is roughly the theory of simple Lie algebra objects in symmetric tensor categories, while the theory of subfactors is roughly that of simple algebra objects in unitary tensor categories. The basic question is then which notions from the classical theory generalize to the quantum setting. For example, is there an analogue of Artin-Wedderburn for semisimple algebra objects in fusion categories? The goal of this post is to argue that the following theorem (due to Ostrik, modulo any errors I’ve introduced) gives a satisfactory generalization.
Any semisimple algebra object in a fusion category is isomorphic (as an algebra object) to the internal endomorphisms End(X) for some object X in a semisimple module category over .
First I’ll unpack the definitions in this statement and then I’ll explain how Artin-Wedderburn for semisimple algebras over a fixed field k follows from this statement. I’ve been thinking about this theorem because Pinhas Grossman and I have been using it to classify “quantum subgroups” of the Haagerup fusion categories, but that’s a story for another day.
I was browsing through different category listings on the arXiv today and noting the changes in numbers of papers over the years. As you might expect, there are more and more papers being posted to the arXiv every year. However, one category defies this trends: QA (quantum algebra).
There are actually less papers being posted to QA in the past three years (2010 317, 2009 308, 2008 323), than there were in the late 90s (1998 364, 1997 434, 1996 395). By contrast, there are about 4 times as many AG papers in the past few years compared to the late 90s, about 10 times as many RT papers, and about 5 times as many GT papers.
What do you make of this? Does it represent a trend in the kind of math that people are doing? Or are people just classifying their work differently?
It would be interesting to see if one can use this arXiv category to get a sense of which fields are becoming more and less popular over time.
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.
- A complete list of possible Frobenius-Perron dimensions in the interval (2, 76/33) for an object in a fusion category.
- 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 ?
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 subfactors. For example, the above identity comes from an equivalence
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.
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.
Update 2: we’ve found a nice answer to our question. Maybe it will appear in the comments soon. –Scott M
Scott, Emily, and I have an ongoing project optimistically called “The Atlas of subfactors.” In the long run we’re hoping to have a site like Dror Bar-Natan and Scott’s Knot atlas with information about subfactors of small index and small fusion categories. In the short run we’re trying to automate known tests for eliminating possible fusion graphs for subfactors.
Right now we’re running into a computational bottleneck: given a number that is a ratio of two algebraic integers how can you quickly test whether it is an algebraic integer? Mathematica’s function AlgebraicIntegerQ is horribly slow, and we’re not sure if that’s because it’s poorly implemented or whether the problem is difficult. So, anyone have a good suggestion? After the jump I’ll explain what this question has to do with tensor categories (and hence subfactors which correspond to bi-oidal categories as I’ve discussed before).
To whet your appetite, here’s an example. Is , where
and where is the largest real root of
an algebraic integer? Mathematica running on Scott’s computer (using the builtin function AlgebraicIntegerQ) takes more than 5 minutes to decide that it is.
Update: Thanks to David Savitt for pointing out that both this example and an earlier one are answered instantly by MAGMA. Blegging is already working. But what’s the trick? Is it something we can teach Mathematica quickly? –Scott M
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 can yield , 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.