Duke Mathematical Journal

Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors

Catharina Stroppel

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To each generic tangle projection from the three-dimensional real vector space onto the plane, we associate a derived endofunctor on a graded parabolic version of the Bernstein-Gel'fand category $\mathcal{O}$. We show that this assignment is (up to shifts) invariant under tangle isotopies and Reidemeister moves and defines therefore invariants of tangles. The occurring functors are defined via so-called projective functors. The first part of the paper deals with the indecomposability of such functors and their connection with generalised Temperley-Lieb algebras which are known to have a realisation via decorated tangles. The second part of the paper describes a categorification of the Temperley-Lieb category and proves the main conjectures of [BFK]. Moreover, we describe a functor from the category of 2-cobordisms into a category of projective functors.

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Duke Math. J. Volume 126, Number 3 (2005), 547-596.

First available in Project Euclid: 11 February 2005

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Stroppel, Catharina. Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors. Duke Math. J. 126 (2005), no. 3, 547--596. doi:10.1215/S0012-7094-04-12634-X. http://projecteuclid.org/euclid.dmj/1108155761.

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