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Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors
Catharina Stroppel
Source: Duke Math. J. Volume 126, Number 3 (2005), 547-596.
Abstract
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 . 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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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1108155761
Digital Object Identifier: doi:10.1215/S0012-7094-04-12634-X
Mathematical Reviews number (MathSciNet):
MR2120117
Zentralblatt MATH identifier:
02154649
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