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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Koszul duality for parabolic and singular category $\mathcal O$
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by Erik Backelin PDF
Represent. Theory 3 (1999), 139-152 Request permission


This paper deals with a generalization of the “Koszul duality theorem” for the Bernstein-Gelfand-Gelfand category $\mathcal O$ over a complex semi-simple Lie-algebra, established by Beilinson, Ginzburg and Soergel in Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473–527. In that paper it was proved that any “block” in $\mathcal O$, determined by an integral, but possibly singular weight, is Koszul (i.e. equivalent to the category of finitely generated modules over some Koszul ring) and, moreover, that the “Koszul dual” of such a block is isomorphic to a “parabolic subcategory” of the trivial block in $\mathcal O$. We extend these results to prove that a parabolic subcategory of an integral and (possibly) singular block in $\mathcal O$ is Koszul and we also calculate the Koszul dual of such a category.
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Additional Information
  • Erik Backelin
  • Affiliation: Department of Mathematics, Albert-Ludwigs-Universitat, Eckerstr. 1, D-79104 Freiburg im Briesgau, Germany
  • Email:
  • Received by editor(s): August 24, 1998
  • Received by editor(s) in revised form: January 31, 1999
  • Published electronically: July 19, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: Represent. Theory 3 (1999), 139-152
  • MSC (1991): Primary 17B10, 18G15, 17B20
  • DOI:
  • MathSciNet review: 1703324