SF&PA – the Temperley-Lieb algebra March 29, 2008Posted by emilypeters in guest post, introductions, planar algebras, small examples, subfactors.
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 points at the bottom of your page, and points at the top, and connect the points up among themselves in a non-crossing way:
We only consider such pictures up to isotopy — then the number of such pictures is exactly the Catalan number (since you can, for instance, read matching parenthesizations as directions for connecting up the points). Now, form a vector space whose basis is Temperley-Lieb pictures on 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:
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 (If there had been circles, we’d have multiplied the picture by ).
If you enjoy multiplying Temperley-Lieb pictures, try this fun exercise: show that Temperley-Lieb is multiplicatively generated by elements , which consist of through strings and a cup and a cap starting at the string:
and satisfy the relations , if and (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 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 diagram, ie a power of :
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 on . Here’s a hard one for you: For which values of 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?