Quadratic duals, Koszul dual functors, and applications

Details Quadratic duals, Koszul dual functors, and applications Quadratic duals, Koszul dual functors, and applications

2006-03-20

arxiv.org/abs/math/0603475v3

Trans. Amer. Math. Soc. 361 (2009), no. 3, 1129-1172

Mathematics

Representation Theory

50 pages, revised version to appear in Trans. AMS

Scientific paper

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.

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