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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 530808, 47 pages
http://dx.doi.org/10.1155/2010/530808
Research Article

A Diagrammatic Temperley-Lieb Categorification

Department of Mathematics, Columbia University, New York, NY 10027, USA

Received 13 March 2010; Revised 26 May 2010; Accepted 14 July 2010

Academic Editor: Alistair Savage

Copyright © 2010 Ben Elias. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 irreducible modules of the Temperley-Lieb algebra.