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.

So suppose we have an additive, -linear category , such that the hom-spaces are finite-dimensional over . Suppose it is pseudo-abelian, i.e. every idempotent has an image. (We don’t want to assume abelianness because a priori we don’t have this for Deligne’s categories.) Finally, suppose the hom-spaces are semisimple algebras. Does it follow that is semisimple?

~~The answer is ~~**no**. Here is a counterexample. We will describe a category . The objects are formal direct sums of two objects . We set , and . Composition is defined in the obvious way on , but we set .

~~It is evident that this is a pseudo-abelian category where the hom-spaces for are semisimple algebras (because can be represented as a sum of copies of and ). However, it is not semisimple. In fact, is not an isomorphism, and it would have to be if were semisimple!~~

OK. So now that we’ve seen what can go horribly wrong, let’s try to develop something that will save us. The problem with was that it didn’t have a *section*: for every , was annihilated. This suggests to us a radical (sorry, this is awful) idea: say that a -linear category is nonnilpotent if for every nonzero there exists some with .

We will show:

Theorem 2 (Knop)Let be nonnilpotent, -linear, pseudo-abelian and suppose the hom-rings are semisimple. Then is semisimple abelian.

I’m basically using the ideas in Sec. 4 of Knop’s paper here but in a modified form; this modification is probably developed somewhere else (such as perhaps Nilpotence, radicaux, et structures monoidales; but I’m not ready to read another heavy paper yet).

So, let’s prove this theorem. The first step is that we can split any into a sum where . This is because there are orthogonal idempotents in the matrix algebra . The point is that the are legit candidates for being simple objects, though it is not immediately obvious.

We shall (following Knop) call any with to be -simple. So we can write any as a direct sum of -simple objects. It turns out that a version of Schur’s lemma still works for these guys:

Lemma 3Let be -simple. Then any map is either zero or an isomorphism.

For if is nonzero, there exists with by nonnilpotence, so is a nonzero multiple of the identity. Thus is left-invertible, and similarly it is right-invertible. It is thus an isomorphism.

So, consider a set of -simple objects which represent them all, i.e. every -simple object is isomorphic to one and only one in . Then any is a direct sum of objects in , in a unique manner.

I now claim that kernels (and by duality, cokernels) exist. But if is -simple, then any morphism

admits a kernel. Namely, we think of the map as a -by- matrix , or equivalently a morphism , take the kernel as a map (or a matrix ) and the associated map . It is then clear that the map

because this is just the sequence of vector spaces

It follows that the sequence

is exact when is a sum of copies of . This is also true when is a sum of copies of other -simple objects (everything’s zero!), so it is exact for all . I.e. we have the kernel of . By taking direct sums, we get kernels in general. And then the category is abelian, so semisimple (the -simple objects are actually simple).

We have proved the theorem.

So, next time, we’ll see how this applies to Deligne’s categories, and what we can do with them.

June 13, 2010 at 2:26 pm

[…] we can do it by appealing to what I discussed in the second post of this series: by proving that the endomorphism rings are semisimple and the category is […]

June 20, 2010 at 1:42 pm

[…] really addressed with a younger version of myself in mind). Readers may wish, however, to review my earlier posts on this […]

January 20, 2011 at 4:36 pm

In your example, End(X+Y) is not semisimple — it contains an obvious nonzero element with square 0. In fact, in a pseudo-abelian category over a field, if End(X) is finite dimensional and semisimple for all X, then the category is semisimple (see, for example, Jannsen 1992).

January 21, 2011 at 1:29 am

Dear mephisto: Sorry, I may be being dense here, but what is wrong with nilpotents in preventing semisimplicity of the endomorphism rings?