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Volodymyr Mazorchuk; Vanessa Miemietz
Serre functors for Lie algebras and superalgebras
(Foncteurs de Serre pour les algèbres de Lie et les super algèbres de Lie)
Annales de l'institut Fourier, 62 no. 1 (2012), p. 47-75, doi: 10.5802/aif.2698
Article PDF | Reviews MR 2986264 | Zbl pre06064511
Class. Math.: 17B10, 16S30, 18G05
Keywords: Lie superalgebra, module, Harish-Chandra bimodule, Serre functor, quiver, category $\mathcal{O}$

Résumé - Abstract

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $\mathcal{O}$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category $\mathcal{O}$ for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra $\mathfrak{q}(2)$.

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