SF&PA: Temperley-Lieb as a planar algebra

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

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,


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:


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:

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?