A Diagrammatic Temperley-Lieb Categorification

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.

long awaited update to published version

References in corpus (3)

• A diagrammatic approach to categorification of quantum groups III
• Highest weight categories arising from Khovanov's diagram algebra II: Koszulity
• The Karoubi envelope and Lee's degeneration of Khovanov homology

Cited by in corpus (2)

• Diagrammatics for Soergel categories
• Trace decategorification of the Hecke category