# Rouquier’s notion of categorification

Here is my first foray into the blogosphere. Unlike Ben, Noah, and Scott, I’m not sufficiently wired for “live conference blogging”, so my first post is going to about a talk I went to almost two months ago.

In mid-May, Nakajima organized a conference on link homology and categorification in Kyoto. There were lots of interesting talks, but the highlight for me was Rouquier’s talks on 2-reps of Lie algebras. I had been anticipating these talks for a long time, since I first heard about his work almost 2 years ago from Sasha Kleshchev. Let me try to explain what he is up to (at least from my perspective).

The starting point is to ask what is the right notion of an action of $sl_2$ on an abelian category. The best way to answer this question is to look at some natural actions of $sl_2$ and then try to see what structure is present. So let us start with an example.

Let $\mathcal{C} = \oplus_n Rep(S_n)$ be the category formed by putting together all the representation categories of $S_n$ (We can do this over any field. In fact, it is most interesting over a finite field — in this case the categories will not be semisimple.) Define a functors $E, F : \mathcal{C} \rightarrow \mathcal{C}$ by one-step induction and restriction respectively. Namely if $V$ is a representation of $S_n$, then $E(V) = Ind_{S_n}^{S_{n+1}} V$.

So what structure do we have? It is easy to see that $E, F$ induce an action of $sl_2$ on the $K(\mathcal{C})$. Also, $E$ has a natural endomorphism, called $X$, given by acting by the group algebra element $(1 n)+ \dots + (n-1 n)$. Also, $E^2$ has a natural endomorphism, called $T$, given by acting by $(n-1 n)$. Such $T$ and $X$ pop up in other settings too, such as category $O$.

Rouquier’s definition of an $sl_2$ categorification on an abelian category $\mathcal{C}$ is the data of two adjoint functors, $E, F$ which generate an action of $sl_2$ on the Grothendieck group, along with an endomorphism $X$ of $E$ and $T$ of $E^2$. $T$ must satisfy a Yang-Baxter equation when action on $E^3$ and $T \circ (1 X) \circ T = q X 1$ acting on $E^2$. A first glance this looks somewhat ad hoc, but using this definition, Chuang and Rouquier are able to prove some strong results. First, with these axioms, the category necessarily splits as a direct sum according to weight spaces in the Grothendieck group and a categorified version of the $EF - FE= H$ identity holds.

But the most impressive part is that they are able prove some classification of representations results. They prove that there is a “minimal” way to category the irreducible representations and then any $sl_2$ categorification comes from these ones (in a somewhat non-trivial way). More spectacularly, they are able to use these ideas to prove some open conjectures in the representation theory of finite groups.

One way to rephrase the above definition is to introduce a monodial category $\mathcal{A}$ which we will be generated by objects $E, F$ and endomorphisms $X, T$ of $E$. Then an $sl_2$ categorification $\mathcal{C}$ will be the same thing as a module category over $\mathcal{A}$ (thanks to David BZ for pointing this out to me). I would like to know if there is some natural definition of $\mathcal{A}$. In particular, is there a geometric definition?

For those who are wondering, there seems to be no relation between this theory and the work I’ve been doing with Sabin. Under Rouquier’s definition, the category breaks into a direct sum corresponding to weight spaces, whereas that doesn’t happen with categories of coherent sheaves.

Rouquier has a similar definition for other Lie algebras, although in those cases the “classification of representations” results are just conjectural. This story is far from finished, but from Rouquier’s results so far, it is clear that there is something very interesting going on.

Also, does this action have anything to do with the Heisenberg action on $\oplus_n Hilb^{[n]} C^2$ ?