## April 14, 2009

### Kamnitzer on Categorifying Tangle Invariants

#### Posted by John Baez

I just read Mike Shulman’s report on talks by people including Tom Leinster and Aaron Lauda at the 88th PSSL in Cambridge — and it’s a funny feeling, because I’m in Glasgow and having fun talking to them here! They sure get around. Together with about 50 other mathematicians, we’re at this workshop:

Every talk at this workshop is worth blogging about, but I won’t have energy for that. Let me talk about the very first one and see what happens from there.

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:

But to understand this stuff you must first understand this:

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.

Posted at April 14, 2009 11:07 AM UTC

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### Re: Kamnitzer on Categorifying Tangle Invariants

Thanks for explaining that. About half-way through you stopped talking about quantum groups and switched to plain-old Lie algebras. How could you have proceeded otherwise? Surely there’s no such thing as an algebraic quantum group. Is the idea to ‘forget’ the deformation, and then reapply it later on after you’ve obtained the Grassmannian?

Posted by: Jamie Vicary on April 14, 2009 3:00 PM | Permalink | Reply to this

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Posted by: Darshan Chande on April 14, 2009 3:56 PM | Permalink | Reply to this

### Re: Kamnitzer on Categorifying Tangle Invariants

those algebraic geometers! […] What’s a derived category? What’s a triangulated category?

While it may be algebraic geometers who like to talk about some of these concepts a lot, I find it useful to remember that many a categorical construction that is widely thought of as being a “tool in algebraic geometry” is not specific to algebraic geometry. It is helpful to know and/or remember that many such concepts just happen to have arisen first and are still mostly studied there for maybe two main reasons:

- Grothendieck happened to have started working in algebraic geometry. Had he started in differential geometry he would have promoted sheaves there and pursued his stacks in that context. No?

- Algebraic test spaces (read: affine algebraic spaces) are more rigid then, say, smooth test spaces. That’s, I think, why in algebraic geometry the need to pass to generalized spaces in terms of sheaves, stacks, $\infty$-stacks on algebraic test spaces arises earlier and with more pressure than in smooth differential geometry: there people keep getting by with faking it and still think of generalized smooth spaces as manifolds of sorts. That may look like an advantage, but is delaying the concept formation here to be as quick as in algebraic geometry.

That’s my impression anyway, please everybody feel free to object.

In any case, I find it worth emphasizing that the notions like derived category and triangulateed category are in no way specific to the context of algebraic geometry, as concepts.

Moreover, there is a nice, deep and explanation for

” …why are […] these concepts beautiful and inevitable?”

Namely: derived triangulated categories are the 1-dimensional shadow of stable $(\infty,1)$-categories.

And the definition of a stable $(\infty,1)$-category is the most obvious and simplest thing in the world, really, despite the possibly scary sounding term. It is certainly way simpler and more transparent than the definition of a triangulated category, even though it gives rise to that definition.

Finally, even the concepts that appear to be intrinsically algebro-geometric, such as derived categories of coherent sheaves, tend to have differential geometric equivalents in the case that one is talking about smooth algebraic spaces, such as those modules for $L_\infty$-algebraic “superconnections” that we talked about #.

For these reasons I feel that it is helpful to disentangle abstract conceptst used in algebraic geometry from their concrete implementation in that context.

But then, maybe I am just missing a crucial point…

Posted by: Urs Schreiber on April 14, 2009 4:07 PM | Permalink | Reply to this

### Re: Kamnitzer on Categorifying Tangle Invariants

Hear, hear.

Posted by: Eugene Lerman on April 14, 2009 5:28 PM | Permalink | Reply to this

### Re: Kamnitzer on Categorifying Tangle Invariants

Eugene Lerman wrote in response to my above comment

Hear, hear.

I have to admit that I don’t know which message is being transported here.

And then I am getting afraid that I missed other messages which were being sent to me between the lines and maybe getting on everybody’s nerves with comments like that.

If so, I apologize. I didn’t mean to come across as obnoxious.

From a long and detailed discussion with Zoran Škoda following up on my message which I had, I do however get the impression that it would be useful to see these issues discussed in a larger circle in public, as it seems to touch on various issues of interest around here, if not on one of the main threads that runs through the $n$-Café’s history.

In the words of David Ben-Zvi:

if not here [on the $n$-Café], where?

Posted by: Urs Schreiber on April 16, 2009 7:16 PM | Permalink | Reply to this

### Re: Kamnitzer on Categorifying Tangle Invariants

Hi Urs,

I was simply agreeing with you statement that

“I find it useful to remember that many a categorical construction that is widely thought of as being a “tool in algebraic geometry” is not specific to algebraic geometry. “

I would add that some things that are specific to algebraic geometry are mixed in with things that are not, and teasing them apart could be quite confusing.

Posted by: Eugene Lerman on April 16, 2009 10:21 PM | Permalink | Reply to this

### Re: Kamnitzer on Categorifying Tangle Invariants

So we have a vector space isomorphism between the intersection cohomology of Gr(λ ) and V(λ ) . Does anyone know if we are able to see the action of the Lie algebra on the cohomology directly?

Posted by: Peter McNamara on April 14, 2009 9:20 PM | Permalink | Reply to this

### Re: Kamnitzer on Categorifying Tangle Invariants

One can see a lot of things (like the weight space decomposition) very easily. To see the full action of the Lie algebra in a direct (rather than Tannakian) fashion you need more work but it is done in

arXiv:math/0005020 E. Vasserot: On the action of the dual group on the cohomology of perverse sheaves on the affine grassmannian

(One should also mention the related series of papers by Feigin Finkelberg Kuznetsov Mirkovic on quasimaps spaces.)

Posted by: David Ben-Zvi on April 14, 2009 10:12 PM | Permalink | Reply to this

### Re: Kamnitzer on Categorifying Tangle Invariants

I wish I had been in Glasgow. Can you say any more? for example,

Which of these varieties can be described some other way? I am interested in spin representations.

Can you say what is an intertwining operator? I assume it is a functor but can you say what invariant means?

Does this categorify the centraliser algebra?

Posted by: Bruce Westbury on April 16, 2009 7:22 PM | Permalink | Reply to this

### Re: Kamnitzer on Categorifying Tangle Invariants

I wish I had been there. Can you say more? For example,

How does taking duals work?

Which of these varieties can be described explicitly? I am interested in the spin representations.

What corresponds to an intertwining operator? I assume it is a functor but what is the condition that says it is invariant?

All this is leading up to: What does this tell us about the centraliser algebra? Do we get a categorification?

Posted by: Bruce Westbury on April 16, 2009 7:30 PM | Permalink | Reply to this

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