Quadratic duals, Koszul dual functors, and applications

Abstract

This paper studies quadratic and duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and duality functors backed up by explicit examples. This generalizes the work of Beilinson, Ginzburg, and Soergel, 1996, in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also de. ne a Koszul duality functor for not necessarily categories. As an illustration of the techniques we reprove the duality (Ryom-Hansen, 2004) of translation and Zuckerman functors for the classical category O in a quite elementary and explicit way. From this we deduce a conjecture of Bernstein, Frenkel, and Khovanov, 1999. As applications we propose a definition of a Koszul dual category for integral blocks of Harish-Chandra bimodules and for blocks outside the critical hyperplanes for the Kac-Moody category O.