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Request: Quivers and Roots November 2, 2008

Posted by David Speyer in Requests, introductions, representation theory.
12 comments

Consider two finite dimensional vector spaces A and B and a linear map \phi between them. Then we can decompose A as K \oplus R where K is the kernel of \phi and R is any subspace transverse to K. Similarly, we can write B as I \oplus C where I is the image of \phi. So we can write \phi as the direct sum of K \to 0, the identity map from R \to I and 0 \to C. At the cost of making some very arbitrary choices, we may simplify even more and say that we can express \phi as the sum of three types of maps: 0 \to k, the identity map k \to k and k \to 0 (where k is our ground field.)

Now, suppose that we have two maps, \phi and \psi from A to B. We’ll start with the case that A and B have the same dimension. If \phi is bijective, then we can choose bases for A and B so that \phi is the identity. Once we have done that, we still have some freedom to change bases further. Assuming that k is algebraically closed, we can use this freedom to put \psi into Jordan normal form. In other words, we can choose bases such that (\phi,\psi) are direct sums of maps like

\left( \left( \begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix} \right), \left( \begin{smallmatrix} \alpha & 1 & 0 \\ 0 & \alpha & 1 \\ 0 & 0 & \alpha \end{smallmatrix} \right) \right).

(Here several different values \alpha may occur in the various summands, and of course, the matrices can be sizes other than 3 \times 3.) If we don’t assume that \phi is bijective (and if we want to allow A and B to have different dimensions) we get a few more cases. But the basic picture is not much worse: in addition to the summands above, we also need to consider the maps

\left( \left( \begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{smallmatrix} \right), \left( \begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix} \right) \right)

(for various sizes n \times (n+1), not just 2 \times 3) and the transpose of these. These three possibilities, and their direct sums, describe all pairs (\phi, \psi) up to isomorphism.

Now, consider the case of three maps. As the dimensions of A and B grow, so do the number of parameters necessary to describe the possible cases. Moreover, almost all cases can not be decomposed as direct sums. More precisely, as long as \mathrm{dim\ } A/\mathrm{dim\ }B is between (3+\sqrt{5})/2 and (3-\sqrt{5})/2, the maps which can be expressed as direct sums of simpler maps have measure zero in the \mathrm{Hom}(A,B). (Where did that number (3+\sqrt{5})/2 come from? Stay tuned!) In the opinion of experts, there will probably never be any good classification of all triples of maps.

The subject of quivers was invented to systemize this sort of analysis. It’s become a very large subject, so I can’t hope to summarize it in one blog post. But I think it is fair to say that anyone who wants to think about quivers needs to start by learning the connection to root systems. So that’s what I’ll discuss here.

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SF&PA: Temperley-Lieb as a planar algebra April 9, 2008

Posted by emilypeters in guest post, introductions, planar algebras, small examples, subfactors.
5 comments

Last week I talked about the Temperley-Lieb algebra – the algebra of boxes with n top points connected in a non-crossing way to n bottom points, with multiplication as stacking boxes.  Some of you may have noticed (but weren’t picky enough to point out) that I didn’t specify whether AB meant A on top of B, or B on top of A.  Of course, it doesn’t really matter, but we should pick one, right?

But wait … why are these two stackings the only candidates for multiplication?  Why shouldn’t we multiply by connecting the right side of a box to the left side of the next box?

or by connecting some top points and some bottom points of each box?  

The observation that there are lots of different multiplications on Temperley-Lieb might lead you to wonder about other operators on Temperley-Lieb.  For instance, we can map TL_n \times TL_n to TL_{2n-1} by connecting any point of the first box to a point of the second, and the rest of the points to the boundary:

Everything I’ve drawn above is an example of a “planar tangle” – and the trace we used last week is also a planar tangle, which takes TL_n to TL_0:

In general, a planar tangle is a diagram where the strings of k input boxes and an output box are connected among themselves in non-crossing ways.  Here’s another example – which is a fine planar tangle, although it’s not clear that it should have any particular meaning if we let it act on Temperley-Lieb inputs.

Planar tangles can sometimes be composed with each other:  we can connect the output of one tangle to the input of another tangle, if both have the same number of strings.  Here’s an unnecessarily complicated example:

Notice that in the LHS, we have labels 1, 2 and 3 in the boxes — this is just so we know what order to do the compositions in.  In the MHS, we’ve stuck the tangles in the boxes they go into; and on the RHS, we’ve discarded the information of the old boundaries and isotoped the strings to make a nicer picture.

The set of planar tangles, with the operation of composition, is an operad.  (I’m not going to tell you what an operad is in general, but if you’re curious  http://homepages.ulb.ac.be/~fschlenk/Maths/What/operad.pdf is a nice introduction.)  A planar algebra is, basically, a representation of the planar operad:  a family of vector spaces with an action by planar tangles which is compatible with composition.  

Temperley-Lieb is not just the simplest and most natural example of a planar algebra; it’s also one of the most important ones.  Coming soon:  Some other examples!

SF&PA – the Temperley-Lieb algebra March 29, 2008

Posted by emilypeters in guest post, introductions, planar algebras, small examples, subfactors.
4 comments

Hi all,

First, I’d like to thank the organizers for inviting me to post on their blog, and apologize for the low tech pictures in what follows.

As Noah mentioned, my name is Emily, I study subfactors and planar algebras, and that’s the back of my head at the top of this page (still). While Noah is taking you through the delights of subfactors sans analysis, I’ll say a few words about planar algebras to set the stage for their later appearance in subfactorland. For now, let’s leave definitions to a future post, and say a little bit about my favorite planar algebra: the Temperley-Lieb algebra.

To get a Temperley-Lieb picture, arrange n points at the bottom of your page, and n points at the top, and connect the points up among themselves in a non-crossing way:

A TL Pic

We only consider such pictures up to isotopy — then the number of such pictures is exactly the n^{th} Catalan number (since you can, for instance, read matching parenthesizations as directions for connecting up the 2n points). Now, form a vector space TL_n whose basis is Temperley-Lieb pictures on 2n points. For instance,

TL3

We turn this vector space into an algebra by defining multiplication: The product of two boxes is the picture you get by stacking them:

TL Multiplication

But what about that loop in the middle? It’s not part of the data of a Temperley-Lieb picture, so we have to throw it out — but let’s remember it was there by multiplying the resulting picture by \delta (If there had been k circles, we’d have multiplied the picture by \delta^k).

If you enjoy multiplying Temperley-Lieb pictures, try this fun exercise: show that Temperley-Lieb is multiplicatively generated by elements e_i, which consist of n-2 through strings and a cup and a cap starting at the i^{th} string:

tlgenerators.jpg

and satisfy the relations e_i^2 = \delta e_i, e_i e_j = e_j e_i if |i-j|>1 and e_i e_{i \pm 1} e_i = e_i (hmm, don’t those last two relations sort of remind you of the braid group?)

One of the reasons we subfactoralists (subfactorers?) like Temperley-Lieb is that it has a lot of structure to it. For instance, we can define an involution ^* on TL_n by horizontal reflection: So, for example:
tlinvolution.jpg

and we can also define a trace by connecting the top points to the bottom points — the result is some number of loops in a TL_0 diagram, ie a power of \delta:

An example of a trace

We call this a trace because it doesn’t care about the order of multiplication (just slide the bottom picture along the strings until it ends up on top).

This combination of a trace and an involution is pretty powerful, as it lets us define a bilinear form \left< x, y \right> := \text{tr}(y^* x) on TL_n. Here’s a hard one for you: For which values of \delta is this form positive definite?

Maybe that’s a good place to stop for now. Coming soon: why is Temperley-Lieb a planar algebra, instead of a just plain algebra?

Group = Hopf algebra October 7, 2007

Posted by Scott Carnahan in Category Theory, crazy ideas, introductions, representation theory.
23 comments

When I was a grad student (not too long ago), my advisor would occasionally get excited about what seemed to me rather minor discoveries. They were often notational efficiencies, or an observation that some extra structure comes for free. Since he was the one with tenure, I figured there was probably a good reason to think that these were important ideas. Eventually, I decided that if you can represent math more efficiently in your head, then you can fit more math into active processing at a time, and you’re more likely to pull something interesting out. This is a big deal if you’re trying to formulate a highly structured argument in a proof, or if you just want to learn some math without wasting large chunks of your life.

Today, I’ll explain one of these ideas, which is that groups and Hopf algebras are really the same thing, even though a lot of people will tell you otherwise.

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Secret Russian Seminar June 10, 2007

Posted by Scott Morrison in introductions.
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I know replacing “Russian” with “Blogging” sounds unpromising, but I thought we should explain the name of the blog!

For a while now (how long?) there’s been a “secret russian seminar” at Berkeley. In fact, it’s not so secret, so much as unannounced. It’s run by, and populated with, the students of various Russian faculty in the Berkeley maths department. In fact, googling for it only produces one hit (if there had been none at all, I’d have kept my mouth shut and not written this post, perhaps). There used to be a secret topology seminar too — perhaps there still is!

Isn’t it hubris to start a blog with no Fields medalists? June 10, 2007

Posted by Noah Snyder in introductions.
2 comments

I’m Noah Snyder, and I was just shocked to realize that not having a Ph.D. puts me in the distinct minority here. I’m a last year student at Berkeley, I’m mostly interested in pictorial algebra and noncommutative tensor products. I also do a bit of old-fashioned representation theory. One could say I study quantum algebra and quantum topology, however since I know so little physics those monikers are a bit misleading.

This summer I’ll be in Denmark for 3 weeks visiting my advisor and attending two conferences at the CTQM, and then I’ll be headed to teach at Mathcamp in Maine. So most of my blogging this summer will be about what I learn at those conferences and the stuff I’m teaching about at Mathcamp. However, I do hope to also put up some posts on what I mean by “pictorial algebra”, a little bit about quantum traces, and possibly some stuff about bimodules over Hopf algebras.

… but will it be the same without beer on the balcony afterwards? June 10, 2007

Posted by Scott Morrison in introductions.
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I’m Scott Morrison, another recent graduate from Berkeley. I’m now working in Santa Barbara, at Microsoft Station Q. I mostly work on Khovanov homology, and mostly on the ‘topological’ aspects of that. Parochial, eh?

But secretly Khovanov homology is another of these messily sprawling subjects, with bits of topology, representation theory, homological algebra, higher category theory, and even algebraic geometry turning up. (But algebraic geometry still scares me, so hopefully Joel will turn up at the blog and be able to look after that aspect of things!)

…and so it begins. June 10, 2007

Posted by Ben Webster in introductions.
6 comments

Welcome, gentle readers!

Let me introduce myself: my name is Ben Webster. I’m a newly minted Ph.D. in mathematics from UC Berkeley, who will be doing a postdoctoral fellowship at the Institute for Advanced Study this year. Since this will put me across the continent from most of the graduate students I’ve chatted with math about over the past 5 years or so (who are slowly being scattered to the four winds as is), I thought it would be nice to carry on our conversations on a blog instead.

Thus, I’m pleased to introduce “Secret Blogging Seminar,” a group blog meant to be something of an extension of the wonderful student seminars we’ve had at Berkeley, hopefully with a bit more input from the outside world.

So, what will we be talking about? Well, math, for starters (with occasional infringements from the rest of the world). My personal research interests sprawl somewhat messily along the the borders between representation theory, algebraic geometry, and knot theory, and different contributors will lean more to one side or another, but I expect the interface between these subjects to be the center of gravity of this blog. I say this not as a restriction, but rather to let you, our valued reader, know what to expect.

Which is not to say that I really know what to expect of this blog. Give me a few weeks and it may be clearer.