I was initially going to talk about why Deligne’s categories of representations of the symmetric group on a nonintegral number of elements are semisimple generically. This is a rather difficult result, and takes quite a bit of preparation in his paper. However, I got sidetracked. Instead, I will devote this post to a general discussion of semisimple categories. According to the material here, it follows that in order to show that Deligne’s categories are semisimple, one has to show that the so-called “partition algebra” is a semisimple ring.
1. Review of semisimple categories
Before we specialize to the case of Deligne’s categories, it may help to go through a little abstract nonsense. Suppose is a semisimple category. This means that is abelian, and each object in is a direct sum of simple objects, where simple means that there is no proper subobject. So for instance, the -modules for a semisimple algebra form a semisimple category. The finite-dimensional representations of a semisimple Lie algebra form a semisimple category (though the finite-dimensional condition is necessary; the enveloping algebra is not a semisimple algebra generally).
Now, I want to look at the hom-spaces in a semisimple category. But first, in the next lemma, there is no need to have the semisimplicity asumption, so I drop that.
Remember Schur’s lemma—that lemma in group representation theory, that any morphism between irreducible representations over is a scalar? The proof of it in different textbooks tends to vary between nonintuitive and clean (depending on the extent of the allegiance of said textbook to category theory). Because when thought of categorically, I claim that it is trivial.
Lemma 1 (Schur, categorical version) Let be a simple object in a -linear abelian category with finite-dimensional hom-spaces. Then . Also, if are simple and nonisomorphic.
So, let’s prove this. We will first prove that any morphism between simple objects is an isomorphism or zero. If one were not zero, it would have either a nontrivial kernel or cokernel. And this would mean either that had a nontrivial subobject or a nontrivial subobject—two things that can’t happen for simple .
It is now clear that when are nonisomorphic, because a nontrivial morphism would be an isomorphism by the above.
Well, then is a ring where every nonzero element is invertible—that is to say, a division algebra. It is also finite-dimensional over by the assumption on the hom-spaces. But every f.d. division algebra over is itself; indeed, if belonged to such a division algebra, then would be a finite extension field (yes, commutative— commutes with itself!) and this cannot happen since is algebraically closed.
In particular, . This proves Schur’s lemma. Not entirely trivial, but at least swift.
So that’s done. I claim then that, in a semisimple category , the hom spaces is ring-isomorphic to a product of matrix algebras over . This is now straightforward: decompose as a sum of simple objects . Partition into equivalence classes based on isomorphism and take the sums of the in each equivalence class. Each has hom-spaces isomorphic to a matrix algebra, so the claim is clear.
In particular, the hom-rings of a semisimple category are—surprise, surprise—semisimple algebras!
2. What if the hom-spaces are semisimple?
The 45-million-dollar question now arises whether the opposite might be true. In fact, I think it is, with certain hypotheses: this isn’t really about Deligne’s paper anymore, but it’s something that I learned from Friedrich Knop’s very interesting paper “Tensor envelopes of regular categories.” Knop actually generalizes Deligne’s construction and axiomatizes it to constructing large classes of interesting tensor categories (such as representation categories of wreath products for finite and complex. I may talk more about Knop’s paper later, but right now I am just using it as a source of some fun abstract nonsense.