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Representation Theory

ISSN 1088-4165



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

Author: Erik Backelin
Journal: Represent. Theory 3 (1999), 139-152
MSC (1991): Primary 17B10, 18G15, 17B20
Published electronically: July 19, 1999
MathSciNet review: 1703324
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Abstract: 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

Received by editor(s): August 24, 1998
Received by editor(s) in revised form: January 31, 1999
Published electronically: July 19, 1999
Article copyright: © Copyright 1999 American Mathematical Society