When confusions annihilate April 13, 2011Posted by Noah Snyder in big list, conferences, planar algebras, quantum algebra, subfactors.
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
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 ?
Extended Haagerup Exists! March 25, 2009Posted by Scott Morrison in conferences, planar algebras, small examples, subfactors, talks.
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 , as long as you change the pivotal structure). The type D planar algebras (with principal graphs the Dynkin diagrams ) were the subject of Noah’s talk at the conference, and the and 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 . He showed that the only possible principal graphs come in two infinite families
(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 March 23, 2009Posted by Noah Snyder in Category Theory, representation theory, subfactors.
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?
TQFTs via Planar Algebras (Part 3) December 10, 2008Posted by Chris Schommer-Pries in low-dimensional topology, Pictorial Algebra, planar algebras, QFT, quantum algebra, subfactors, talks, tqft.
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) December 6, 2008Posted by Chris Schommer-Pries in low-dimensional topology, Pictorial Algebra, planar algebras, QFT, subfactors, talks, tqft.
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 December 5, 2008Posted by Chris Schommer-Pries in low-dimensional topology, Pictorial Algebra, planar algebras, QFT, subfactors, talks, tqft.
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 November 26, 2008Posted by Noah Snyder in fun problems, group theory, linear algebra, subfactors.
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…
Bleg: testing algebraic integrality by computer. October 13, 2008Posted by Noah Snyder in blegs, Category Theory, knot atlas, Number theory, quantum algebra, subfactors, things I don't understand, Uncategorized.
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