The sl_3 web algebra
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. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLRalgebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus}_0(\mathcal{W}^S)_{\mathbb{Q}(q)}$, 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2012
 arXiv:
 arXiv:1206.2118
 Bibcode:
 2012arXiv1206.2118M
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Geometric Topology
 EPrint:
 Numbering matched with the published version, no other changes