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Quadratic duals, Koszul dual functors, and applications


Authors: Volodymyr Mazorchuk, Serge Ovsienko and Catharina Stroppel
Journal: Trans. Amer. Math. Soc. 361 (2009), 1129-1172
MSC (2000): Primary 16S37, 18E30, 16G20, 17B67
DOI: https://doi.org/10.1090/S0002-9947-08-04539-X
Published electronically: October 8, 2008
MathSciNet review: 2457393
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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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Additional Information

Volodymyr Mazorchuk
Affiliation: Department of Mathematics, Uppsala University, Box 480, 751 06, Uppsala, Sweden
Email: mazor@math.uu.se

Serge Ovsienko
Affiliation: Department of Mathematics, Kyiv University, 64, Volodymyrska st., 01033, Kyiv, Ukraine
Email: ovsko@voliacable.net

Catharina Stroppel
Affiliation: Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, United Kingdom
Email: cs@maths.gla.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-08-04539-X
Received by editor(s): April 26, 2006
Published electronically: October 8, 2008
Additional Notes: The first author was partially supported by the Swedish Research Council
The second author was partially supported by the Royal Swedish Academy of Sciences and The Swedish Foundation for International Cooperation in Research and Higher Education (STINT)
The third author was supported by The Engineering and Physical Sciences Research Council (EPSRC)
Article copyright: © Copyright 2008 American Mathematical Society

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