Thomas Gobet (Kaiserslautern), A categorification of the Temperley-Lieb algebra by analogues of Soergel bimodules.

Abstract:

We consider a set of subvarieties of the geometric representation of the symmetric group on which the Kauffman monoid acts. Using regular functions on these varieties we introduce an additive graded category having analogues of Soergel bimodules as objects. We also define as product a slightly different operation than a usual tensor product, which is not associative in general as product of bimodules, but becomes associative on the subcategory which we are interested in. The Temperley-Lieb algebra turns out to be isomorphic to the split Grothendieck ring of that category. The image of the canonical diagram basis under this isomorphism is the set of classes of indecomposable bimodules and one can associate to the left and right annihilators of any indecomposable bimodule two varieties which carry out all the information about the corresponding Temperley-Lieb diagram.