[Foncteurs de Serre pour les algèbres de Lie et les super algèbres de Lie]
Nous proposons une nouvelle réalisation du foncteur de Serre pour la catégorie 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 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 (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 .
We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category 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 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 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 .
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 $\mathcal{O}$
Mazorchuk, Volodymyr 1 ; Miemietz, Vanessa 2
@article{AIF_2012__62_1_47_0,
author = {Mazorchuk, Volodymyr and Miemietz, Vanessa},
title = {Serre functors for {Lie} algebras and superalgebras},
journal = {Annales de l'Institut Fourier},
pages = {47--75},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {62},
number = {1},
year = {2012},
doi = {10.5802/aif.2698},
mrnumber = {2986264},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2698/}
}
TY - JOUR AU - Mazorchuk, Volodymyr AU - Miemietz, Vanessa TI - Serre functors for Lie algebras and superalgebras JO - Annales de l'Institut Fourier PY - 2012 SP - 47 EP - 75 VL - 62 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2698/ DO - 10.5802/aif.2698 LA - en ID - AIF_2012__62_1_47_0 ER -
%0 Journal Article %A Mazorchuk, Volodymyr %A Miemietz, Vanessa %T Serre functors for Lie algebras and superalgebras %J Annales de l'Institut Fourier %D 2012 %P 47-75 %V 62 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2698/ %R 10.5802/aif.2698 %G en %F AIF_2012__62_1_47_0
Mazorchuk, Volodymyr; Miemietz, Vanessa. Serre functors for Lie algebras and superalgebras. Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 47-75. doi : 10.5802/aif.2698. https://aif.centre-mersenne.org/articles/10.5802/aif.2698/
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