Categorification of the Kazhdan-Lusztig basis of the Temperley-Lieb algebra by bimodules

TL;DR

This paper introduces a categorification of the Temperley-Lieb algebra using bimodules similar to Soergel bimodules, with a novel monoidal structure different from the standard tensor product.

Contribution

It provides a new categorification framework for the Temperley-Lieb algebra employing bimodules with a unique monoidal operation.

Findings

Realization of the Temperley-Lieb algebra via bimodules

Development of a non-standard monoidal structure

Potential applications to representation theory

Abstract

We realize the Temperley-Lieb algebra by analogues of Soergel bimodules. The key point is that the monoidal structure is not given by a usual tensor product but by a slightly more complicated operation.