×

Unicity of graded covers of the category \(\mathcal{O}\) of Bernstein-Gelfand-Gelfand. (English) Zbl 1394.17021

Summary: We show that the standard graded cover of the well-known category \(\mathcal{O}\) of Bernstein-Gelfand-Gelfand can be characterized by its compatibility with the action of the center of the enveloping algebra.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
16W50 Graded rings and modules (associative rings and algebras)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bernstein, J. N., Gelfand, I. M. and Gelfand, S. I., A certain category of \(\mathfrak{g} \)-modules, Funktsional. Anal. i Prilozhen10:2 (1976), 1-8 (Russian). English transl.: Funct. Anal. Appl.10 (1976), 87-92.
[2] Beilinson, A. A. and Ginsburg, V., Mixed categories, Ext-duality and representations (results and conjectures), Preprint, 1986. · Zbl 0980.17003
[3] Beilinson, A. A., Ginsburg, V. and Soergel, W., Koszul duality patterns in representation theory, J. Appl. Math. Stoch. Anal.9 (1996), 473-527. · Zbl 0864.17006
[4] Gabriel, P., Des catégories abéliennes, Bull. Soc. Math. France90 (1962), 323-448. · Zbl 0201.35602
[5] König, S., Slungård, I. H. and Xi, C., Double centralizer properties, dominant dimension, and tilting modules, J. Algebra240 (2001), 393-412. · Zbl 0980.17003 · doi:10.1006/jabr.2000.8726
[6] Liu, Q., Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, Oxford, 2009.
[7] Rottmaier, M., Diploma thesis, University of Freiburg, Freiburg, 2012. · Zbl 1426.20017
[8] Riche, S., Soergel, W. and Williamson, G., Modular Koszul duality, Compos. Math.150 (2014), 273-332. · Zbl 1426.20017 · doi:10.1112/S0010437X13007483
[9] Soergel, W., Kategorie 𝒪, Perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc.3 (1990), 421-445. · Zbl 1114.83314
[10] Soergel, W., The combinatorics of Harish-Chandra bimodules, J. Reine Angew. Math.429 (1992), 49-74. · Zbl 1323.32001
[11] Soergel, W., On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra152 (2000), 311-335. · Zbl 1101.14302 · doi:10.1016/S0022-4049(99)00138-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.