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 KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K_0^{\oplus }({\mathcal{W}}^S{)}_{\mathbb{Q}(q)}$, 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.

Numbering matched with the published version, no other changes

References in corpus (2)

• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Graded decomposition numbers for cyclotomic Hecke algebras

Cited by in corpus (21)

• Webs and quantum skew Howe duality
• The Blanchet-Khovanov algebras
• Deformations of colored sl(N) link homologies via foams
• Integral presentations of quantum lattice Heisenberg algebras
• Cellular structures using ${\text{U}}_q$-tilting modules
• Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)
• An odd categorification of quantum sl(2)
• Functoriality of colored link homologies
• Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality
• Khovanov homology for virtual tangles and applications
• sl(N)-Web categories
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• The sl(N)-web algebras and dual canonical bases
• Relative cellular algebras
• A characterisation of indecomposable web-modules over Khovanov-Kuperberg Algebras
• Generic ${\mathfrak{gl}}_2$-foams, web and arc algebras
• Anchored foams and annular homology
• Braid group actions from categorical symmetric Howe duality on deformed Webster algebras
• Tensor diagrams and Chebyshev polynomials
• Real Springer fibers and odd arc algebras
• A formula to evaluate type A webs and link polynomials