Quadratic Duals, Koszul Dual Functors, and Applications
2009 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 361, no 3, 1129-1172 p.Article in journal (Refereed) Published
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 de. ne 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 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.
Place, publisher, year, edition, pages
2009. Vol. 361, no 3, 1129-1172 p.
IdentifiersURN: urn:nbn:se:uu:diva-106981ISI: 000261255300001OAI: oai:DiVA.org:uu-106981DiVA: diva2:227523