In some previous posts, I’ve written about the relationship between categorification and geometry. I’m banging on this because I think it’s an underappreciated point, especially because a lot of people right now are coming at categorification with a background in topology and higher algebra, and thus aren’t as familiar with the geometric and representation theoretic techniques that actually underlie a lot of, say, what I do.
In particular, a lot of people (myself among them) are very excited about the categorification of quantum groups at the moment and there are two different ways of approaching said categorification:
- You can write down a bunch of algebras and categories in terms of generators and relations. This is the approach used in the work of Chuang and Rouquier and of Khovanov and Lauda. It has a lot of advantages: there are lots of parameters one can tweak (which of course, leads to the issue of different authors tweaking them differently), it’s very generally applicable, and it doesn’t really require any machinery one doesn’t acquire in a graduate algebra class, aside from a little terminology.
- You can take a geometric approach and work with varieties of quiver representations. This work was pioneered by Lusztig, though it has seen some very interesting developments in recent years in the work of Zheng and Li. This approach requires some very deep technology, involving perverse sheaves, etc. and is less flexible (in particular, it’s still not properly understood how to extend it to the case of non-symmetric Cartan matrix). On the other hand, geometry gives us powerful tools to show that things that are supposed to be non-zero or positive really are. In particular, certain properties of canonical bases require some quite deep geometric theorems.
The point I want to make in this post is that these approaches are the same. Of course, “the same” is a very vague term in mathematics, but here, I think it’s justified. For example, consider the following theorems:
Theorem. (Lusztig) The Grothendieck group of the category of perverse sheaves on the moduli stack of representations of a Dynkin quiver is for the positive unipotent part of the corresponding Lie algebra.
Theorem. (Khovanov-Lauda/Rouquier) There exists a 2-category whose monoidal category of representations has Grothendieck group .
These theorems live in different worlds (unfortunately, the former only works for simply-laced groups; the latter works for all). But, actually, they are the same.
Theorem. (Vasserot-Varagnolo/Rouquier) The derived category of the moduli stack of representations of a Dynkin quiver is equivalent to the category of dg-modules over (considered as a dg-category with trivial differential).
Furthermore, this might sound like a horrifying thing to prove, in fact it’s pretty easy. You just have to be careful about your language (I’m going to say everything in terms of stacks, but secretly I’m picking a basis for all my representations, and working equivariantly with respect to change of basis). The most important arguments go like this:
- One can realize all the perverse sheaves on the moduli space of quiver reps using pushforward from quiver flag varieties, moduli spaces of quiver representations with a fixed flag. By usual homological algebra, one can understand the whole derived (dg-)category of the moduli stack by computing the Ext-spaces between these pushforwards (in dg-land).
- Chriss and Ginzburg tell us that we can think of this as the Borel-Moore homology of the moduli space of representations equipped with pairs of the flags, using the usual convolution.
- Write down elements of the Borel-Moore homology which correspond to the generators of the 2-morphisms in the category . One of these is a dot on the ith strand from the left; that goes to multiplication by the Chern class of the tautological line bundle for the ith step of the flag. The other is a crossing; that goes to the fundamental class of the space of reps with a pair of flags, one corresponding to the top sequence, the other corresponding to the bottom sequence, and agreeing except at the index where the crossing happens.
- Check these satisfy the right relations: this is easiest to see from how they act on the cohomology of the spaces of moduli with flags (these are exactly the “polynomial representation” of ).
- Check that these generate: this is easily done by taking a filtration with respect to relative positions of flags, and seeing that these still generate afterwards.
So, strikingly, these stories that maybe at first glance looked like they were in different worlds are really the same. With more work one can also start to match up the 2-categories corresponding to the whole algebra from the work of Khovanov-Lauda and of Rouquier with geometric 2-categories as defined by Li, but that story is very far from fully worked out.
As usual, having two perspectives on the same story is a very powerful thing. I personally would never have been able to make heads or tails of the categorification story if I hadn’t had the geometry to guide me; there are a lot of arguments in my papers on the subject I just would not have seen without geometric insight. On the other hand, geometry is hard. I tried for the better part of a year to construct categorified quantum invariants using geometry (and I still believe it is possible), before switching to the algebraic approach because it is so much more flexible. It’s very advantageous thing in mathematics to have two stories in one’s head about the same thing, and I hope this blog post helps speed the intermingling of these two approaches.