SF&PA: From Subfactors to Planar Algebras

So, suppose you have a Subfactor category. Remember that this is a bi-oidal category with four different kinds of objects (A-A, A-B, B-A, and B-B), with tensor products \otimes_A and \otimes_B, with a good theory of duals, and a chosen generating object X. How does one study categories with tensor products? By drawing pictures!

The usual yoga for this is that objects are drawn using lines labeled with that object, morphisms are little boxes, tensor products are disjoint unions, and duals are the same line but with the orientation the other way. For example, here is some morphism f \otimes g: V\otimes W \rightarrow W \otimes U \otimes U^*:

The fact that there’s a nice theory of duals means that you can draw more complicated pictures like the following one where f \in Hom(V \otimes V, V^* \otimes V^*) and the whole picture lives in Hom(V^{\otimes 4}, 1).

But how should we modify these pictures so that they’re best adapted to the Subfactor setup? First we need to be able to keep track of which P-Q sector various objects are living, and second we want to emphasize the tensor generator X.

There’s a very natural way to keep track of which sector objects live in by just shading the regions of the pictures. That is an A-B object would be written with an unshaded region on the left and a shaded region on the right. It could thus only be tensor-producted with another object which had a shaded region on its left (so that the shadings match up). Thus you can tensor product it with a B-A or B-B object, just as you’d hope.

The way to emphasize the importance of X is to just only allow strands labeled by X! Since every object is in some tensor product of X’s and X*’s if we just know the Hom-spaces between such tensor products we can recover everything by just identifying some other object V, say its in X \otimes X^* with the projection onto V in End(X \otimes X^*).

But now notice that X and X* are shaded differently (one is an A-B object and the other is a B-A object), so we don’t need to orient the strands since the shading tells us which are X’s and which are X*’s. So in the end all we need to think about are pictures that look like the following (which is interpreted as an element of Hom(X \otimes X^* \otimes X \rightarrow X)):

These pictures can be put together in two obvious ways: composition (vertical stacking) and tensor product (horizontal disjoint union). As Emily explained in her last post though they can be composed in lots of more interesting ways that fit together into what Vaughan Jones calls a “Planar Algebra.” The next step in the story is to not only forget about the original Subfactor, but to also forget about the Subfactor category, and only remember the Planar Algebra we get out at the end. But for now, there’s a Warriors game on, so I should get going.

One thought on “SF&PA: From Subfactors to Planar Algebras

Comments are closed.