\(S\)-subcategories in \(O\)
type: Item
instance of: scholarly article
title: \(S\)-subcategories in \(O\)
zbMATH Open document ID: 1018.17005
author: Volodymyr Mazorchuk
author: Vyacheslav Futorny
author: Steffen Koenig
published in: Manuscripta Mathematica
publication date: 2003-09-11T00:00:00Z
review text: In 1976 \textit{I. N. Bernstein, I. M. Gel'fand} and \textit{S. I. Gel'fand} introduced the category \(\mathcal{O}\) for a complex semisimple Lie algebra \(\mathfrak{g}\) [Sel. Math. Sov. 1, 121-142 (1981); translation from Tr. Semin. Im. I. G. Petrovskogo 2, 3-21 (1976; Zbl 0499.22004)]. Two basic properties were established there, namely the block decomposition of \(\mathcal{O}\) (these blocks being equivalent to a module category over a quasi-hereditary algebra) and the BGG reciprocity principle between Verma, projective and simple modules. In this paper the authors analyze subcategories of \(\mathcal{O}\) consisting of complete modules admitting quasi-Verma flags. It is proven that the subcategories are admissible in the sense used in another paper by the the authors [Algebr. Represent. Theory 5, 259-276 (2002; Zbl 1031.17008)]. It is proven that these categories are Abelian (this property is not inherited by the category \(\mathcal{O}\)). A combinatorial description for these subcategories analogous to that existing for \(\mathcal{O}\) is obtained. Using the double centralizer property of Soergel, \(\mathcal{S}\)-subcategories are introduced, and new proofs of the character formula for the tilting module of \(\mathcal{O}\) [\textit{W. Soergel}, Represent. Theory 2, 432-448 (1998; Zbl 0964.17018)] and the self-duality of Ringel for the principal block of \(\mathcal{O}\) are obtained. The proof use the Enright completion functor [\textit{T. J. Enright}, Ann. Math. (2) 110, 1-82 (1979; Zbl 0417.17005)].
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16G10
zbMATH DE Number: 1542484
zbMATH Keywords: Verma flags
zbMATH Keywords: Ringel self-duality
zbMATH Keywords: double centralizer property
reviewed by: Rutwig Campoamor Stursberg
MaRDI profile type: Publication
full work available at URL: s002290070038
OpenAlex ID: W2033751112
DOI: 10.1007/S002290070038
description: scientific article; zbMATH DE number 1542484