Diagrammatic and geometric categorification June 21, 2011

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:

1. 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.
2. 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. (more…)