# The Unapologetic Mathematician

## Friday, Carnies, and Khovanov homology

The new Carnival of Mathematics post is up over at Science and Reason, continuing my efforts to become the blogosphere’s go-to guy for publicizing the Atlas project.

This afternoon in the graduate student seminar, Joshua Sussan broke down parts of the famous paper of Bernstein, Frenkel, and Khovanov that launched knot homology theories: A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak{sl}_2)$ via projective and Zuckerman functors. Yes, Zuckerman is my advisor, and yes this also ties back into the stuff I’ve been talking about with respect to $E_8$.

One particular bit of self-promotion on this point: often Khovanov homology — either this original representation-theoretic approach, a later sheaf-theoretic approach, or Khovanov and Rozansky’s combinatoral version — is called a categorification of the Jones polynomial, or of the bracket polynomial. I’ve mentioned the bracket before, specifically in relation to my talk on bracket extensions. In fact, Khovanov homology on tangles categorifies one of my bracket-extending functors: $F_{V_1,\cap}$, where $V_1$ is the standard $2$-dimensional representation of the $q$-deformed enveloping algebra $U_q(\mathfrak{sl}_2)$ and $\cap$ is the canonical pairing from $V_1\otimes V_1$ to the trivial representation.

Don’t get me wrong. Khovanov homology is a truly brilliant idea, but I hold out hope that there’s some other categorification that makes it clear what the topological content of the Kauffman bracket polynomial is.

## Tensor products of abelian groups

Often enough we’re going to see the following situation. There are three abelian groups — $A$, $B$, and $C$ — and a function $f:A\times B\rightarrow C$ that is linear in each variable. What does this mean?

Well, “linear” just means “preserves addition” as in a homomorphism, but I don’t mean that $f$ is a homomorphism from $A\times B$ to $C$. That would mean the following equation held:
$f(a+a',b+b')=f(a,b)+f(a',b')$

Instead, I want to say that if we fix one variable, the remaining function of the other variable is a homomorphism. That is
$f(a,b+b')=f(a,b)+f(a,b')$
$f(a+a',b)=f(a,b)+f(a',b)$
$f(a+a',b+b')=f(a,b)+f(a,b')+f(a',b)+f(a',b')$
we call such a function “bilinear”.

The tensor product is a construction we use to replace bilinear functions by linear functions. It is defined by yet another universal property: a tensor product of $A$ and $B$ is an abelian group $T$ and a bilinear function $t:A\times B\rightarrow T$ so that for every other bilinear function $f:A\times B\rightarrow C$ there is a unique linear function $\bar{f}:T\rightarrow C$ so that $f(a,b)=\bar{f}(t(a,b))$. Like all objects defined by universal properties, the tensor product is automatically unique up to isomorphism if it exists.

So here’s how to construct one. I claim that $T$ has a presentation by generators and relations as an abelian group. For generators take all elements of $A\times B$, and for relations take all the elements $(a+a',b)-(a,b)-(a',b)$ and $(a,b+b')-(a,b)-(a,b')$ for all $a$ and $a'$ in $A$ and $b$ and $b'$ in $B$.

By the properties of such presentations, any function of the generators — of $A\times B$ — defines a linear function on $T$ if and only if it satisfies the relations. That is, if we apply a function $f$ to each relation and get ${}0$ every time, then $f$ defines a unique function $\bar{f}$ on the presented group. So what does that look like here?
$f(a+a',b)-f(a,b)-f(a',b)=0$
$f(a,b+b')-f(a,b)-f(a,b')=0$
So a bilinear function $f$ gives rise to a linear function $\bar{f}$, just as we want.

Usually we’ll write the tensor product of $A$ and $B$ as $A\otimes B$, and the required bilinear function as $(a,b)\mapsto a\otimes b$.

Now, why am I throwing this out now? Because we’ve already seen one example. It’s a bit trivial now, but catching it while it’s small will help see the relation to other things later. The distributive law for a ring says that multiplication is a bilinear function of the underlying abelian group! That is, we can view a ring as an abelian group $R$ equipped with a linear map $m:R\otimes R\rightarrow R$ for multiplication.

April 6, 2007

## Zuckerman on KLV (preview)

Even though Andrew Wiles was speaking at the Branford College master’s tea today, I didn’t go. Zuckerman was giving the first of two or three lectures on this whole KLV thing. And I actually took notes!

Unfortunately the scanner in the computer lab was being evil, so I can’t post them quite yet. I’ll definitely have them by Monday, though, and after that I’ll try to explain what I wrote. They should be already more than readable to mathematicians, though.

What I do have is pictures of a conceptual diagram we constructed on the blackboard in his office the other day. I managed to get it mostly into three parts: 1 2 3 (~700KB each). I apologize for the quality of the middle one — I couldn’t use a flash without washing out the board entirely. It should still be readable. The third cuts off some lists of names associated with the topics they’re next to. The second list is “Jantzen, Vogan, Speh”, while in the first list “H.C.” is Harish-Chandra and “Z.” is Zuckerman.

As for what all this means, that’s what these lectures are to explain more thoroughly. Here we see the entire subject circling “characters”, which are certain functions on the groups we’re interested in. Properly defining them was the subject of today’s lecture. In the lower left is a list of examples of the sorts of groups we’re interested in — $E_8,8(\mathbb{R})$ is the now-(in)famous one. To the right of the diagram is the statement that two special classes of representations, the “standard” and “irreducible” ones, are related in a certain way. On the right is the recipe for computing the irreducible representations into which the Atlas project’s computation fits.

April 6, 2007