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

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 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.