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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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
  • MR Author ID: 353912
  • Email:
  • Serge Ovsienko
  • Affiliation: Department of Mathematics, Kyiv University, 64, Volodymyrska st., 01033, Kyiv, Ukraine
  • Email:
  • Catharina Stroppel
  • Affiliation: Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, United Kingdom
  • Email:
  • 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)
  • © Copyright 2008 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 1129-1172
  • MSC (2000): Primary 16S37, 18E30, 16G20, 17B67
  • DOI:
  • MathSciNet review: 2457393