## Koszul duality for parabolic and singular category $\mathcal O$

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- by Erik Backelin
- Represent. Theory
**3**(1999), 139-152 - DOI: https://doi.org/10.1090/S1088-4165-99-00055-2
- Published electronically: July 19, 1999
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## Abstract:

This paper deals with a generalization of the â€śKoszul duality theoremâ€ť 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), 473â€“527. In that paper it was proved that any â€śblockâ€ť 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 â€śKoszul dualâ€ť of such a block is isomorphic to a â€śparabolic subcategoryâ€ť 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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## Bibliographic Information

**Erik Backelin**- Affiliation: Department of Mathematics, Albert-Ludwigs-Universitat, Eckerstr.Â 1, D-79104 Freiburg im Briesgau, Germany
- Email: erik@toto.mathematik.uni-freiburg.de
- 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: https://doi.org/10.1090/S1088-4165-99-00055-2
- MathSciNet review: 1703324