The $\mathfrak{sl}_3$ web algebra Title: The $\mathfrak{sl}_3$ web algebra Authors: Marco Mackaay, Weiwei Pan and Daniel Tubbenhauer Status: Math. Z. 277-1-2 (2014), 401-479. Last update: Sun, 10 Nov 2013 12:38:22 GMT ArXiv link: http://arxiv.org/abs/1206.2118 ArXiv version = 0.99 published version, but beware that the numbering differs 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: If $w_2$ and $w_3$ are different, then, by convention, the multiplication is zero. If $w_2=w_3$, then glue the foams together via a multiplication foam. 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. 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. 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. NEWS I am still a fool. My paper got accepted. The arXiv version of this paper was updated. My paper got accepted. The arXiv version of this paper was updated. "There are two ways to do mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else - but persistent." - based on a quotation from Raoul Bott. Upcoming event where you can meet me: Visit Faro Click
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