The paper studies quadratic and Koszul 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 Koszul duality functors backed up by explicit examples. This generalises previous results in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also define a ``Koszul'' duality functor for not necessarily Koszul categories. As an illustration of the techniques we reprove the Koszul duality of translation and Zuckerman functors for the classical category O in a quite elementary and explicit way. 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.
CITATION STYLE
Mazorchuk, V., Ovsienko, S., & Stroppel, C. (2008). Quadratic duals, Koszul dual functors, and applications. Transactions of the American Mathematical Society, 361(03), 1129–1172. https://doi.org/10.1090/s0002-9947-08-04539-x
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