The n-Category Café

On Monday April 13th, Joel Kamnitzer gave the first of three talks on Categorification and Algebra. More specifically, his talk was about categorifying some famous invariants of tangles coming from quantum groups — the so-called ‘Reshetikhin–Turaev invariants’. His approach to categorifying these amounts to replacing certain vector spaces (representations of quantum groups) by certain categories (derived categories of coherent sheaves on certain varieties). This approach has not been fully worked out except in some special cases, but it’s already tremendously charming. Here are some papers on it:

• Joel Kamnitzer and Sabin Cautis, Knot homology via derived categories of coherent sheaves I, sl(2) case.
• Joel Kamnitzer and Sabin Cautis, Knot homology via derived categories of coherent sheaves II, sl(m) case.

But to understand this stuff you must first understand this:

• Nicolai Reshetikhin and Vladimir Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1-26.

This is one of those papers that has launched a thousand ships. Until you understand it — or one of the many subsequent retellings of the story it tells — a vast amount of post-1990 mathematics, including much of what I write about, will seem obscure and mysterious. So, let me quickly summarize it, before saying a word about how Kamnitzer and Cautis are trying to categorify it!

Every semisimple Lie algebra $\mathbf{g}$ gives a Hopf algebra $U_q \mathbf{g}$ called a ‘quantum group’, or more precisely a ‘$q$-deformed universal enveloping algebra’. The main nice thing about this quantum group is that it has a category of finite-dimensional representations that’s very much like that of $\mathbf{g}$ itself, but is braided monoidal instead of symmetric monoidal. Indeed, this category — let’s call it $C$ — is a braided monoidal category with duals, so it gives tangle invariants.

Why? Well, Shum’s theorem says that the category of tangles is the free braided monoidal category with duals on one object. So, if we pick any object in $C$, say $V$, any tangle with $m$ strands coming in and $n$ strands going out gives us a morphism from $V^{\otimes m}$ to $V^{\otimes n}$. And this is our tangle invariant!

More generally, we can take any tangle with strands labelled by objects in $C$, and get a morphism from some tensor product of objects to some other tensor product of objects. This invariant of labelled tangles is called a Reshetikhin–Turaev invariant.

In fact, everything I’ve said so far is very general: $C$ could be any braided monoidal category with duals. To ‘categorify’ this story means to replace $C$ by a braided monoidal 2-category with duals, say $\tilde{C}$, which reduces to $C$ when we decategorify it. Doing this is again a very general sort of challenge. But to do this when $C$ is the category of representations of the quantum group $U_q \mathbf{g}$ is a challenge whose solution seems to involve lots of special features of quantum groups. And that’s where things get really interesting.

How can we tackle this challenge? I’ll just describe the first step: namely, how to ‘categorify’ a representation of $U_q \mathbf{g}$, obtaining a derived category of coherent sheaves on something like a flag variety. In what sense does this ‘categorify’ the representation we started with? Well, a derived category of coherent sheaves is an example of a triangulated category. We can take the $K$-theory of any triangulated category and get a vector space. And, if we take the $K$-theory of this particular triangulated category we get back our representation.

“Hey, wait a minute!” you cry. “You’re starting to sound like one of them — those algebraic geometers! Aren’t you going to explain all these buzzwords? You’re supposed to be the guy who actually explains stuff! What’s a flag variety. What’s a coherent sheaf? What’s a derived category? What’s a triangulated category? How do you take the $K$-theory of such a thing? And more importantly, why are all these concepts beautiful and inevitable?”

Alas, these are things I am not allowed to explain. If you become an algebraic geometer, and pay your initiation fees, someone will knock on your door one night, take you out back, and whisper a few sentences in your ear. Then all these concepts will become incredibly clear! But having been initiated myself, and taken the vow of secrecy, I am not at liberty to say these sentences in public. Someday I may rebel. But not yet.

So: let me follow the usual practice, and suddenly assume (without any justification) that you’ve already undergone the initiation and understand all the buzzwords.

To specify a finite-dimensional irreducible representation of the quantum group $U_q \mathbf{g}$, or for that matter of the Lie algebra $\mathbf{g}$, I just need to specify a dominant integral weight $\lambda$. This gives a finite-dimensional irrep $V(\lambda)$. More generally any list of dominant integral weights $\lambda_1, \dots, \lambda_n$ specifies a representation $V(\lambda_1) \otimes \cdots \otimes V(\lambda_n)$ of the quantum group. And this is the sort of representation we want to categorify.

How do we do it?

Well, we use something called the Geometric Satake Correspondence.

There’s a semisimple algebraic group $\check{G}$ called the Langlands dual of $G$, where $G$ is the simply-connected algebraic group whose Lie algebra is $\mathbb{g}$. The Langlands dual has the following wondrous property. It has a maximal torus, say $\check{T}$, such that

$$hom(\mathbb{C}^*, \check{T})$$

is naturally isomorphic to the weight lattice of $G$.

Now let’s form the ‘affine Grassmannian’ of $\check{G}$. This is

$$Gr = \check{G}(K)/\check{G}(O)$$

where $O$ is the field of formal power series in one variable and $K$ is the field of formal Laurent series. This is an infinite-dimensional gadget which plays a major role in Pressley and Segal’s book Loop Groups. You should think of it as a Grassmannian for the loop group of $\check{G}$ — it’s the Grassmannian associated to the ‘extra dot’ in the Dynkin diagram for this loop group. It’s a union of finite-dimensional Grassmannians $Gr(\lambda)$ where $\lambda$ ranges over all dominant integral weights of $\mathbf{g}$.

I believe that Joel restricted to a special case of this story right about now, to protect us from a buzzword we hadn’t paid enough to understand: perverse sheaves. Namely, he focused attention on representations $V(\lambda_i)$ that are ‘miniscule’, meaning that all the weights in the weight decomposition lie in the same Weyl group orbit. In this case each $Gr(\lambda_i)$ is a smooth projective variety. Smoothness protects us from the need to understand perverse shaeaves. And in this case he described how to build a smooth projective variety $Gr(\lambda_1, ... ,\lambda_n)$ that’s a sort of like a flag variety: it’s a $Gr(\lambda_n)$ bundle over a $Gr(\lambda_{n-1})$ bundle over a… get the pattern?… over $Gr(\lambda_1)$.

I realize now that I don’t have the energy to precisely describe this construction… but let me explain the point of it:

Geometric Satake Correspondence (Lusztig et al) — the cohomology $H_*(Gr(\lambda_1, ... , \lambda_n))$ is isomorphic to the representation $V(\lambda_1) \otimes \cdots \otimes V(\lambda_n)$.

And, since $Gr(\lambda_1, ... \lambda_n)$ is smooth, we can think of this cohomology as the $K$-theory of the derived category of coherent sheaves on $Gr(\lambda_1, ... , \lambda_n)$. So, in this sense, the derived category of coherent sheaves on $Gr(\lambda_1, ... , \lambda_n)$ categorifies the representation $V(\lambda_1) \otimes \cdots \otimes V(\lambda_n)$.

Of course this is just an isomorphism of vector spaces so far; we need much more to categorify the whole braided monoidal category of representations of this sort! But that’s enough for now — I’ve already missed lunch. No time to explain how this stuff actually sort of makes sense to me, at least in special cases.