2012/06/30 by Marco Mackaay, M. Mackaay, W. Pan +3 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Arc (geometry) #Center (category theory) #Cohomology #Cohomology ring #Grothendieck group #Ring (chemistry) #math.GT #math.QA
paper · pdf · doi:10.1007/s00209-013-1262-6
published as Math. Z. 277:1-2 (2014), 401-479 · Numbering matched with the published version, no other changes
openalex publication_date 2013/12/14 · openalex created_date 2016/06/24 · arxiv created 2018/03/10 · arxiv updated 2018/03/13 · openalex updated_date 2026/08/05
In this paper we use Kuperberg's \mathfraksl3-webs and Khovanov's \mathfraksl3-foams to define a new algebra KS, which we call the \mathfraksl3-web algebra. It is the \mathfraksl3 analogue of Khovanov's arc algebra. We prove that KS is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that KS is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K⊕0(WS)ℚ(q), to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that KS is a graded cellular algebra.