A Diagrammatic Temperley-Lieb Categorification

TL;DR

This paper introduces a diagrammatic categorification of the Temperley-Lieb algebra using planar graphs, connecting algebraic structures with topological and combinatorial models.

Contribution

It defines a new quotient category of Soergel bimodules that categorifies the Temperley-Lieb algebra through planar graph representations.

Findings

Establishes a planar graph-based categorification of the Temperley-Lieb algebra.

Links ideals in the quotient to Coxeter complex topology and 2D cobordisms.

Outlines how subquotients categorify cell modules of the algebra.

Abstract

The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the cell modules of the Temperley-Lieb algebra.