Koszul duality for parabolic

and singular category

Author:
Erik Backelin

Journal:
Represent. Theory **3** (1999), 139-152

MSC (1991):
Primary 17B10, 18G15, 17B20

DOI:
https://doi.org/10.1090/S1088-4165-99-00055-2

Published electronically:
July 19, 1999

MathSciNet review:
1703324

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with a generalization of the ``Koszul duality theorem'' for the Bernstein-Gelfand-Gelfand category 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 , 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 .

We extend these results to prove that a parabolic subcategory of an integral and (possibly) singular block in is Koszul and we also calculate the Koszul dual of such a category.

**[AJS]**H.H. Andersen, J.C. Jantzen and W. Soergel,*Representations of quantum groups at a th root of unity and of semisimple groups in characteristic : independence of*, Astérisque 220 (1994), 3-321. MR**95j:20036****[BG]**J. Bernstein and S.I. Gelfand,*Tensor products of finite and infinite dimensional representations of semisimple Lie algebras*, Comp. Math. 41 (1981), 245-285. MR**82c:17003****[BGG1]**J. Bernstein, I.M. Gelfand and S.I. Gelfand,*Category of -modules*, Functional Anal. Appl. 10 (1976), 87-92.**[BGG2]**J. Bernstein, I.M. Gelfand and S.I. Gelfand,*Schubert cells and cohomology of spaces*, Russian Math. Survey 28 (1973), no. 3, 87-92.**[BGS]**A. Beilinson, V. Ginzburg and W. Soergel,*Koszul duality patterns in representation theory*, J. Amer. Math. Soc. 9 (1996), 473-527. MR**96k:17010****[CPS]**E. Cline, B. Parshall and L. Scott,*Abstract Kazhdan-Lusztig theories*, Tohoku Math. J. 2, Ser. 45, No. 4, (1993), 511-534. MR**94k:20079****[Jan]**J. C. Jantzen,*Einhüllende Algebren halbeinfacher Lie-algebren*, Springer-Verlag (1983). MR**86c:17011****[KL1]**D. Kazhdan and G. Lusztig,*Tensor structures arising from affine Lie algebras, I, II*, J. Amer. Math. Soc. 6 (1993), 905-1011. MR**93m:17014****[KL2]**D. Kazhdan and G. Lusztig,*Tensor structures arising from affine Lie algebras, III, IV*, J. Amer. Math. Soc. 7 (1994), 335-453. MR**94g:17048**; MR**94g:17049****[Soe1]**W. Soergel,*Kategory , perverse Garben und Moduln über den Koinvarianten Algebra zur Weylgruppe*, J. Amer. Math. Soc. 2 (1990), 421-445. MR**91e:17007****[Soe2]**W. Soergel,*-Cohomology of simple highest weight modules on walls and purity*, Invent. Math. 98 (1989), 565-580. MR**90m:22037**

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Additional Information

**Erik Backelin**

Affiliation:
Department of Mathematics, Albert-Ludwigs-Universitat, Eckerstr. 1, D-79104 Freiburg im Briesgau, Germany

Email:
erik@toto.mathematik.uni-freiburg.de

DOI:
https://doi.org/10.1090/S1088-4165-99-00055-2

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