Category \({\mathcal O}\): gradings and translation functors.
type: Item
instance of: scholarly article
title: Category \({\mathcal O}\): gradings and translation functors.
zbMATH Open document ID: 1040.17002
DOI: 10.1016/S0021-8693(03)00308-9
author: Catharina Stroppel
published in: Journal of Algebra
publication date: 2003-10-22T00:00:00Z
review text: The author presents a systematic approach to the construction of a graded version of the Bernstein-Gelfand-Gelfand category \(\mathcal{O}\) from [\textit{I. N. Bernshtein}, \textit{I. M. Gelfand} and \textit{S. I. Gelfand}, Category of \(\mathfrak{g}\) modules, Funct. Anal. Appl. 10, 87--92 (1976; Zbl 0353.18013)]. It has been shown in [\textit{A. Beilinson}, \textit{V. Ginzburg} and \textit{W. Soergel}, Koszul duality patterns in representation theory, J. Am. Math. Soc. 9, 473--527 (1996; Zbl 0864.17006)] that a block of the category \(\mathcal{O}\) is equivalent to the category of finite dimensional (not necessarily graded) modules over some graded (even Koszul) algebra. Thus it makes sense to speak about gradable and non-gradable modules in \(\mathcal{O}\), gradable and non-gradable endofunctors, etc. In the present paper it is shown that the usual duality on \(\mathcal{O}\) and the translation functors on \(\mathcal{O}\) are gradable. Using this a lot of gradable modules are constructed. Many classical exact sequences involving Verma modules or dual Verma modules are shown to be gradable, and many classical multiplicity formulae are shown to have graded analogues. At the same time, the author also constructs an example of a module in \(\mathcal{O}\) which does not admit a graded lift. The author also describes the relation between the graded combinatorics of \(\mathcal{O}\) and the Hecke algebra.
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 1995432
zbMATH Keywords: category \(\mathcal{O}\)
zbMATH Keywords: Koszul algebra
zbMATH Keywords: graded algebra
zbMATH Keywords: graded module
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1995432