Stroppel, Catharina Category \({\mathcal O}\): gradings and translation functors. (English) Zbl 1040.17002 J. Algebra 268, No. 1, 301-326 (2003). The author presents a systematic approach to the construction of a graded version of the Bernstein-Gelfand-Gelfand category \(\mathcal{O}\) from [I. N. Bernshtein, I. M. Gelfand and S. I. Gelfand, Category of \(\mathfrak{g}\) modules, Funct. Anal. Appl. 10, 87–92 (1976; Zbl 0353.18013)]. It has been shown in [A. Beilinson, V. Ginzburg and 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. Reviewer: Volodymyr Mazorchuk (Uppsala) Cited in 2 ReviewsCited in 36 Documents MSC: 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 16W50 Graded rings and modules (associative rings and algebras) Keywords:category \(\mathcal{O}\); Koszul algebra; graded algebra; graded module Citations:Zbl 0353.18013; Zbl 0864.17006 PDFBibTeX XMLCite \textit{C. Stroppel}, J. Algebra 268, No. 1, 301--326 (2003; Zbl 1040.17002) Full Text: DOI References: [1] Andersen, H.; Jantzen, J.; Soergel, W., Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\), Astérisque, 220 (1994) · Zbl 0802.17009 [2] Bass, H., Algebraic K-theory (1968), Benjamin · Zbl 0174.30302 [3] Beilinson, A.; Bernstein, J. N., Localisation de \(g\)-modules, C. R. Acad. Sci. Paris, 292, 15-18 (1981) · Zbl 0476.14019 [4] Beilinson, A.; Bernstein, J. N., A proof of Jantzen conjectures, Adv. 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