The $$\mathfrak {sl}_{3}$$ 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 KLR-algebra 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.