TQFT with corners and tilting functors in the Kac-Moody case

Abstract

We study projective functors (i.e. direct summands of compositions of translations through walls) for parabolic versions of $\cO$ as well as for integral regular blocks outside the critical hyperplanes in the symmetrizable Kac-Moody case. It turns out that in both situations the functors are completely determined by their restriction to the additive category generated by (the limit of) a `full projective tilting' object. We describe how projective functors in the parabolic setup give rise to an invariant of tangle cobordisms and formulate a conjectural direct connection to Khovanov homology. Our main result, however, is the classification theorem for indecomposable projective functors in the Kac-Moody case verifying a conjecture of F. Malikov and I. Frenkel.

References in corpus (7)

• Khovanov's homology for tangles and cobordisms
• A functor-valued invariant of tangles
• An invariant of link cobordisms from Khovanov homology
• Categorification of the braid groups
• Koszul duality of translation--and Zuckerman functors
• Projective-injective modules, Serre functors and symmetric algebras
• Perverse sheaves, Koszul IC-modules, and the quiver for the category O

Cited by in corpus (9)

• The Blanchet-Khovanov algebras
• Integral presentations of quantum lattice Heisenberg algebras
• Highest weight categories arising from Khovanov's diagram algebra III: category O
• An invariant of tangle cobordisms via subquotients of arc rings
• Generic ${\mathfrak{gl}}_2$-foams, web and arc algebras
• Witt groups of abelian categories and perverse sheaves
• Category O for quantum groups
• Braid cobordisms, triangulated categories, and flag varieties
• On selfadjoint functors satisfying polynomial relations