Serre functors for Lie algebras and superalgebras

Mazorchuk, Volodymyr; Miemietz, Vanessa

Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 47-75.

Nous proposons une nouvelle réalisation du foncteur de Serre pour la catégorie $𝒪$ de BGG associée à une algèbre de Lie semi-simple complexe de dimension finie, en utilisant les bimodules d’Harish-Chandra. De plus, nous démontrons que dans beaucoup de cas notre réalisation s’applique aux super algèbres de Lie classiques. Pour cela, nous prouvons que la catégorie $𝒪$ et ses généralisations paraboliques pour les super-algèbres de Lie classiques sont des catégories avec foncteurs pleins projectifs. Comme application, nous montrons que, dans beaucoup de cas, l’algèbre d’endomorphismes du module projectif-injectif basique de la catégorie $𝒪$ (parabolique) pour les super-algèbres de Lie est symétrique. En particulier, dans ce cas, les algèbres décrivant les blocs de la catégorie de modules de dimension finie sont symétriques. Nous calculons ces dernières algèbres pour la super algèbre de Lie $𝔮\left(2\right)$.

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $𝒪$ 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 $𝒪$ 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 $𝒪$ 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 $𝔮\left(2\right)$.

Classification : 17B10, 16S30, 18G05
Keywords: Lie superalgebra, module, Harish-Chandra bimodule, Serre functor, quiver, category $\mathcal{O}$
Mot clés : super algèbres de Lie, bimodules d’Harish-Chandra, foncteur de Serre, carquois, catégorie $\mathcal{O}$

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