By a subfactor I will mean a pair of rings A < B, such that (and I apologize in advance if I mess this up, as I said I don’t know any analysis):
- A and B are both Von Neumann algebras
- A and B have trivial centers (that is they are factors)
- A and B are both II_1-factors (that is they have a unique normalized trace)
- A<B has finite index (perhaps I’ll define this later)
- A<B is irreducible (that is the centralizer of A in B is trivial)
Whew! What on earth does all of this mean? And why would a representation theorist care about these conditions?
Well, as we saw before in the Galois theory example, given a pair A<B we get a bi-oidal category C(A<B) which is tensor generated by and . It turns out that you can use the above conditions to prove a lot of nice algebraic properties about C(A<B). Furthermore, due to a converse theorem of Ocneanu Popa’s given a suitable category C(A<B) it must come from a subfactor. With this (hard!) theorem in hand we can happily ignore the subfactor setup and instead just think about categories that look like C(A<B).
Furthermore, if category theory scares you, tomorrow we’ll be doing the first thing you should do when someone tries to talk to you about categories: translate everything into pictures! At that point you’ll be able to forget both the Subfactor and the category and just start drawing pictures.
The fact that A and B are both Von Neumann algebras has two important consequences. First, all the Hom-spaces in this category are not only vector spaces over , they also inherit a positive definite Hermitian form. Secondly, the analysis allows one to prove (and this is substantially harder) that C(A<B) is a semisimple category. Morally this isn’t surprising, as positive definite forms are often helpful for proving semi-simplicity, but the proof is far from obvious.
The fact that A and B are factors says that the trivial bimodules and are irreducible. The fact that A<B is irreducible says that the generating modules and are also irreducible.
Finiteness of the index tells you that there is a good dimension theory for C(A<B). The II_1 property together with finiteness of the index (and maybe irreduciblity?) imply that C(A<B) has a good theory of duals.
Let’s unpack this last bit for a moment. When we were talking about Galois theory, C(L/K) had the property that had a unique copy of the trivial bimodule (and this “evaluation map” is just given by multiplication in L). So, behaves like the dual of . However, on the other hand if you took the tensor product in the opposite order it didn’t work out so well. That is, we didn’t have a unique map In the Subfactor world, there is a unique A-A bilinear map that is called the “conditional expectation.”
To summarize, we’ll call a bi-oidal category C (bi-oidal means that every object comes with two labels A-A, A-B, B-A, or B-B, and that we have two associative tensor products and which are defined when the labels match up) together with a choice of object X in the A-B sector a subfactor category if:
- C is enriched over Hilbert spaces (that is every Hom space has a Hilbert space structure)
- C is semi-simple
- C has a really nice theory of duals. (The right condition here is “spherical” in the sense of Barrett and Westbury, modified appropriately for the bi-oidal setting. Included in this condition is the fact that C has a good notion of dimensions.)
- The trivial objects and are irreducible
- X is irreducible, furthermore X is a tensor generator in the sense that every object in C occurs as a sub-object of or .
One thing I want to emphasize here is that the choice of X is part of the data of a Subfactor category.