Data

1. Title: $\mathfrak{sl}_n$-webs, categorification and Khovanov-Rozansky homologies
2. Author: Daniel Tubbenhauer
3. Status: Preprint. Last update: Mon, 21 Jul 2014 15:42:54 GMT
5. LaTex Beamer presentation: Slides1, Slides2, Slides3

Abstract

In this paper we define an explicit basis for the $\mathfrak{sl}_n$-web algebra $H_n(\vec{k})$, the $\mathfrak{sl}_n$ generalization of Khovanov's arc algebra $H_{2}(m)$, using categorified $q$-skew Howe duality.
Our construction, which can be seen as a $\mathfrak{sl}_n$-web version of Hu and Mathas graded cellular basis for the cyclotomic KL-R algebras of type $A$, has two major applications. The first is that it is a graded cellular basis. The second is that it can be explicitly computed for any $\mathfrak{sl}_n$-web $w=v^*u$ which gives a basis of the corresponding $2$-hom space between $u$ and $v$. We use this fact to give a (in principle) computable version of Khovanov-Rozansky $\mathfrak{sl}_n$-link homology. The complex we define for this purpose can be realized in the KL-R setting and needs only $F$'s and no $E$'s.
Moreover, we discuss some application of our construction on the uncategorified level related to dual canonical bases of the $\mathfrak{sl}_n$-web space $W_n(\vec{k})$ and the MOY-calculus. Latter gives rise to a method to compute colored Reshetikhin-Turaev $\mathfrak{sl}_n$-link polynomials.

A few extra words

The main observation is that the $\mathfrak{sl}_n$-web space $W_n(\vec{k})$ is, under $q$-skew Howe duality, the $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-weight module of weight $\vec{k}$ in a certain $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-highest weight module.
It turns out that every $\mathfrak{sl}_n$-web can be obtained by a string of only $F$'s acting as elements of $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$ on a highest weight vector. Thus, we can say that the $\mathfrak{sl}_n$-web space $W_n(\vec{k})$ is an instance of $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-highest weight theory.
A closed $\mathfrak{sl}_n$-web is therefore nothing else than a quantum number since one jumps from the $1$-dimensional highest weight space to the $1$-dimensional lowest weight space using the $F$'s of $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$.
As a consequence: One can write each (colored) link as string of $F$'s together with operators $T$ for crossings. As for example displayed below for the Hopf link (with restriction to $n=2$). Here we act with the $F$'s from bottom to top and the $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-weight spaces can be read of from the number grid.
The operators $T$ measure the difference how one can hop around from the highest to the lowest weight and we obtain the Reshetikhin-Turaev $\mathfrak{sl}_n$-link polynomials therefore as an instance of $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-highest weight theory.
History repeats itself: The same is true for the $\mathfrak{sl}_n$-link homologies. We just replace the weight modules with weight categories, the $F$'s with functors and the measurement of the difference with certain natural transformations.
And since everything an the level of the $\mathfrak{sl}_n$-link polynomials takes place in a certain $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-module of highest weight we can obtain the Khovanov-Rozansky $\mathfrak{sl}_n$-link homologies by only using the cyclotomic KL-R algebra.
Thus, these homologies are instances of categorified $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-highest weight theory.