Data

1. Title: The $\mathfrak{sl}_3$ web algebra
2. Authors: Marco Mackaay, Weiwei Pan and Daniel Tubbenhauer
3. Status: Math. Z. 277-1-2 (2014), 401-479. Last update: Sat, 10 Mar 2018 12:15:16 GMT
5. ArXiv version = 0.99 published version
6. LaTex Beamer presentation: Slides1

Abstract

In this paper we use Kuperberg's $\mathfrak{sl}_3$ webs and Khovanov's $\mathfrak{sl}_3$ foams to define a new algebra $K_S$, which we call the $\mathfrak{sl}_3$ web algebra. It is the $\mathfrak{sl}_3$ analogue of Khovanov's arc algebra $H(n)$. We prove that $K_S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of quantum skew Howe duality, which allows us to prove that $K_S$ is Morita equivalent to a certain, cyclotomic KLR-algebra. This allows us to determine the Grothendieck group of $K_S$, to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K_S$ is a graded cellular algebra.

A few extra words

The main idea of the $\mathfrak{sl}_3$ web algebra is simple, i.e. given four webs $w_1,w_2,w_3$ and $w_4$, which are the same at the boundary, one can define two new webs $w_1w_2$ and $w_3w_4$ by rotating $w_1$ and $w_3$ around the x-axis, switching their orientation and glue them on top of $w_2$ and $w_4$. Then a multiplication $m(f(w_1w_2),f(w_3w_4))$ of foams with the corresponding webs as boundary can be defined as follows:
1. If $w_2$ and $w_3$ are different, then, by convention, the multiplication is zero.
2. If $w_2=w_3$, then glue the foams together via a multiplication foam.
3. The multiplication foam is made of saddles (arc-split), unzips (Y-split) and square removals (H-split). The picture shows a saddle and an unzip.
4. The picture below is one part of the proof that the multiplication does not depend on the isotopy type. Note that the part of the foams corresponding to $w_1$ and $w_4$ is not shown, since it will be the identity.
5. Note that one can speak of a categorification, i.e. on web level the multiplication is trivial (erase the middle and glue top and bottom), but on foam level lots of interesting mathematics is happening.