A generalization of the category $ \mathcal{O}$ of Bernstein–Gelfand–Gelfand

Guillaume Tomasini

doi:10.1016/j.crma.2010.03.008

Lie Algebras

A generalization of the category

O

of Bernstein–Gelfand–Gelfand
[Une généralisation de la catégorie

O

de Bernstein–Gelfand–Gelfand]

Présenté par : Michel Duflo

Guillaume Tomasini ¹

¹ IRMA, CNRS et Université de Strasbourg, 7, rue René-Descartes, 67084 Strasbourg cedex, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 509-512.

Résumé
Abstract

L'étude des représentations irréductibles d'une algèbre de Lie simple définie sur le corps des nombres complexes a conduit Bernstein, Gelfand et Gelfand a introduire une catégorie qui fournit un cadre naturel pour les modules de plus haut poids. Le but de cette note est de présenter une construction d'une famille de catégories généralisant celle de Bernstein–Gelfand–Gelfand. Nous décrivons les modules simples de certaines de ces catégories. Cette classification permet de montrer que ces catégories sont semi-simples.

In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this note, we define a family of categories which generalizes the BGG category. We classify the simple modules for some of these categories. As a consequence we show that these categories are semisimple.

Reçu le : 2009-12-16
Accepté le : 2010-03-12
Publié le : 2010-03-29

DOI : 10.1016/j.crma.2010.03.008

Affiliations des auteurs :

Guillaume Tomasini ¹

¹ IRMA, CNRS et Université de Strasbourg, 7, rue René-Descartes, 67084 Strasbourg cedex, France

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     author = {Guillaume Tomasini},
     title = {A generalization of the category $ \mathcal{O}$ of {Bernstein{\textendash}Gelfand{\textendash}Gelfand}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {509--512},
     publisher = {Elsevier},
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     number = {9-10},
     year = {2010},
     doi = {10.1016/j.crma.2010.03.008},
     language = {en},
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Guillaume Tomasini. A generalization of the category $ \mathcal{O}$ of Bernstein–Gelfand–Gelfand. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 509-512. doi : 10.1016/j.crma.2010.03.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.008/

Bibliographie
Cité par

[1] G. Benkart; D. Britten; F. Lemire Modules with bounded weight multiplicities for simple Lie algebras, Math. Z., Volume 225 (1997), pp. 333-353

[2] I. Bernšteĭn, I. Gelfand, S. Gelfand, Differential operators on the base affine space and a study of $g$ -modules, in: Lie Groups and Their Representations, Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971, 1975, pp. 21–64

[3] N. Bourbaki Groupes et Algèbres de Lie, Chap. IV–VI, Hermann, Paris, 1968

[4] D. Britten; F. Lemire A classification of simple Lie modules having a 1-dimensional weight space, Trans. Amer. Math. Soc., Volume 299 (1987), pp. 683-697

[5] D. Britten; F. Lemire Tensor product realizations of simple torsion free modules, Canad. J. Math., Volume 53 (2001), pp. 225-243

[6] D. Britten; V. Futorny; F. Lemire Submodule lattice of generalized Verma modules, Comm. Algebra, Volume 31 (2003), pp. 6175-6197

[7] D. Britten; O. Khomenko; F. Lemire; V. Mazorchuk Complete reducibility of torsion free $C_{n}$ -modules of finite degree, J. Algebra, Volume 276 (2004), pp. 129-142

[8] A. Coleman; V. Futorny Stratified L-modules, J. Algebra, Volume 163 (1994), pp. 219-234

[9] Y. Drozd; V. Futorny; S. Ovsienko Harish-Chandra subalgebras and Gelfand–Zetlin modules, Finite-Dimensional Algebras and Related Topics, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 424, 1994, pp. 79-93

[10] S. Fernando Lie algebra modules with finite-dimensional weight spaces. I, Trans. Amer. Math. Soc., Volume 322 (1990), pp. 757-781

[11] V. Futorny, The weight representations of semisimple finite dimensional Lie algebras, PhD thesis, Kiev University, 1987

[12] V. Futorny; V. Mazorchuk Structure of α-stratified modules for finite-dimensional Lie algebras. I, J. Algebra, Volume 183 (1996), pp. 456-482

[13] V. Futorny; V. Mazorchuk Highest weight categories of Lie algebra modules, J. Pure Appl. Algebra, Volume 138 (1999), pp. 107-118

[14] V. Futorny; S. König; V. Mazorchuk A combinatorial description of blocks in $O (P, Λ)$ associated with $sl (2)$ -induction, J. Algebra, Volume 231 (2000), pp. 86-103

[15] V. Futorny; S. König; V. Mazorchuk $S$ -subcategories in $O$ , Manuscripta Math., Volume 102 (2000), pp. 487-503

[16] V. Futorny; S. König; V. Mazorchuk Categories of induced modules and standardly stratified algebras, Algebr. Represent. Theory, Volume 5 (2002), pp. 259-276

[17] V. Futorny; A. Molev; S. Ovsienko The Gelfand–Kirillov conjecture and Gelfand–Tsetlin modules for finite W-algebras, Adv. Math., Volume 223 (2010), pp. 773-796

[18] D. Grantcharov; V. Serganova Cuspidal representations of $sl (n + 1)$ | arXiv

[19] J. Humphreys Representations of Semisimple Lie Algebras in the BGG Category $O$ , Grad. Stud. Math., vol. 94, American Mathematical Society, Providence, RI, 2008

[20] N. Jacobson Basic Algebra II, W.H. Freeman, 1980

[21] O. Mathieu Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble), Volume 50 (2000), pp. 537-592

[22] V. Mazorchuk Generalized Verma Modules, Math. Stud. Monogr. Ser., vol. 8, VNTL Publishers, L'viv, 2000

[23] V. Mazorchuk; S. Ovsienko Submodule structure of generalized Verma modules induced from generic Gelfand–Zetlin modules, Algebr. Represent. Theory, Volume 1 (1998), pp. 3-26

[24] V. Mazorchuk; C. Stroppel Cuspidal ${sl}_{n}$ -modules and deformations of certain Brauer tree algebras | arXiv

[25] I. Penkov; V. Serganova Generalized Harish-Chandra modules, Mosc. Math. J., Volume 2 (2002) no. 4, pp. 753-767

[26] A. Rocha-Caridi Splitting criteria for $g$ -modules induced from a parabolic and the Berňsteĭn–Gelfand–Gelfand resolution of a finite-dimensional, irreducible $g$ -module, Trans. Amer. Math. Soc., Volume 262 (1980), pp. 335-366

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