Quadratic duals, Koszul dual functors, and applications

Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina

doi:10.1090/S0002-9947-08-04539-X

Quadratic duals, Koszul dual functors, and applications
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by Volodymyr Mazorchuk, Serge Ovsienko and Catharina Stroppel PDF

Trans. Amer. Math. Soc. 361 (2009), 1129-1172 Request permission

Abstract:

This 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 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 define a “Koszul” duality functor for not necessarily Koszul categories. As an illustration of the techniques we reprove the Koszul duality (Ryom-Hansen, 2004) of translation and Zuckerman functors for the classical category $\mathcal {O}$ 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 $\mathcal {O}$.

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