### Categorified Quantum Groups

#### Posted by David Corfield

Once in a distant blog, John was quick to pour cold water on the suggestion I made that the aims of those categorifying might differ sufficiently to merit distinguishing types of ‘categorification’:

I don’t like this “Frenkelian” versus “Baezian” distinction. Baez was inspired to work on higher categories thanks to the work of Crane and Frenkel. Frenkel’s student Khovanov cites Baez’s work on 2-tangles in his first paper on categorified knot invariants. Frenkel’s student Khovanov has taken on Baez’s student Lauda as a postdoc at Columbia starting next fall. Will their work on categorifying quantum groups and using these to get 2-tangle invariants be “Frenkelian” or “Baezian”?

Some results of the collaboration are now out. Aaron has just posted A categorification of quantum sl(2) to the arXiv.

Did the Geometric Representation Theory Seminar reach the point of having categorified quantum groups? Not that I’m after differences of approach, of course.

Posted at March 27, 2008 6:34 PM UTC
## Re: Categorified Quantum Groups

Aaron starts with:

Personally, while I am acquainted with quantum groups, we have not become friends yet. Since their representation categories may coincide, I like to think of quantum groups as a certain incarnation of centrally extended loop groups.

These, in turn, can be thought of as being essentially Lie 2-groups, as described in From loop groups to 2-groups.

From that point of view, there is no question that you eventually want to categorify this.

I would have to spend probably a lot of time to see if the following is related to what Aaron is considering, but let me say it anyway:

the centrally extended loop groups are entirely controlled by a 3-cocycle on the underlying group. This becomes most strikingly manifest when we pass to the corresponding Lie 2-groups and then their Lie 2-algebras: these are the “String type” Lie 2-algebras $g_\mu$ described by John and Alissa in HDA VI which are

entirelydetermined by a Lie algebra and a 3-cocycle on it.So probably we want to be looking at higher generalizations of that.

In

$L_\infty$-connections# we observe that there are actually large families of “higher String-like” Lie algebras of this kind:there is a Lie $n$-algebra $g_\mu$ for every Lie $k$-algebra with an Lie $(n+1)$-cocycle on it. If this cocycle is in transgression with an invariant polynomial, this is of “String type” and generalized the String Lie 2-algebra which connects to centrally extended loop groups.

One would hence expect that integrating these higher String-type Lie $n$-algebras up to $n$-groups and looking at their representation theory relates to the categorification that Aaron is after.

(Every such Lie $n$-algebra can be integrated to an $\omega$-group as I describe for instance at the end of space and quantity.)

Just an observation. I haven’t even started trying to think about whether this has any relevance for what Aaron is doing.