The linear $\protect \mathfrak{n}(1|N)$–invariant differential operators and $\protect \mathfrak{n}(1|N)$–relative cohomology

Hafedh Khalfoun; Ismail Laraiedh

doi:10.5802/crmath.22

Algèbres de Lie, Physique mathématique

The linear

𝔫 (1 | N)

–invariant differential operators and

𝔫 (1 | N)

–relative cohomology
[Opérateurs différentiels linéaires

𝔫 (1 | N)

–invariants et cohomology

𝔫 (1 | N)

–relative]

Hafedh Khalfoun ^1,² ; Ismail Laraiedh ^1,²

¹ Departement of Mathematics, College of Sciences and Humanities - Kowaiyia, Shaqra University, Kingdom of Saudi Arabia
² Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie

Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 45-58.

Résumé
Abstract

Sur le supercercle $(1, N)$ -dimensionnel $S^{1 | N}$ , nous classifions les opérateurs différentiels linéaires $𝔫 (1 | N)$ -invariant agissant sur les densités tensorielles sur $S^{1 | N}$ , où $𝔫 (1 | N)$ est la superalgèbre de Lie de Heisenberg. Ce résultat permet de calculer le premier espace de cohomologie différentiels $𝔫 (1 | N)$ -relative de la superalgèbre de Lie des champs de vecteurs de contact $𝒦 (N)$ à coefficients dans le superespace des densités tensorielles. Pour $N = 0, 1, 2,$ nous etudions le premier espace de cohomologie $𝔫 (1 | N)$ -relative de $𝒦 (N)$ dans le superespace de l’algèbre supercommutative $𝒮𝒫 (N)$ des symboles pseudodifférentiels sur $S^{1 | N}$ et dans la superalgèbre de Lie $𝒮 Ψ 𝒟 𝒪 (S^{1 | N})$ des opérateurs superpseudodifférentiels. Nous donnons explicitement les 1-cocycles engendrent ces espaces de cohomologie.

Over the $(1, N)$ -dimensional supercircle $S^{1 | N}$ , we classify $𝔫 (1 | N)$ -invariant linear differential operators acting on the superspaces of weighted densities on $S^{1 | N}$ , where $𝔫 (1 | N)$ is the Heisenberg Lie superalgebra. This result allows us to compute the first differential $𝔫 (1 | N)$ -relative cohomology of the Lie superalgebra $𝒦 (N)$ of contact vector fields with coefficients in the superspace of weighted densities. For $N = 0, 1, 2,$ we investigate the first $𝔫 (1 | N)$ -relative cohomology space associated with the embedding of $𝒦 (N)$ in the superspace of the supercommutative algebra $𝒮𝒫 (N)$ of pseudodifferential symbols on $S^{1 | N}$ and in the Lie superalgebra $𝒮 Ψ 𝒟 𝒪 (S^{1 | N})$ of superpseudodifferential operators with smooth coeffcients. We explicity give 1-cocycles spanning these cohomology spaces.

Reçu le : 2019-06-19
Révisé le : 2019-08-07
Accepté le : 2020-01-06
Publié le : 2020-03-18

DOI : 10.5802/crmath.22

Classification : 53D55, 14F10, 17B10, 17B68

Affiliations des auteurs :

Hafedh Khalfoun ^{1, 2} ; Ismail Laraiedh ^{1, 2}

¹ Departement of Mathematics, College of Sciences and Humanities - Kowaiyia, Shaqra University, Kingdom of Saudi Arabia
² Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2020__358_1_45_0,
     author = {Hafedh Khalfoun and Ismail Laraiedh},
     title = {The linear $\protect \mathfrak{n}(1|N)${\textendash}invariant differential operators and $\protect \mathfrak{n}(1|N)${\textendash}relative cohomology},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {45--58},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {1},
     year = {2020},
     doi = {10.5802/crmath.22},
     language = {en},
}

TY  - JOUR
AU  - Hafedh Khalfoun
AU  - Ismail Laraiedh
TI  - The linear $\protect \mathfrak{n}(1|N)$–invariant differential operators and $\protect \mathfrak{n}(1|N)$–relative cohomology
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 45
EP  - 58
VL  - 358
IS  - 1
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.22
LA  - en
ID  - CRMATH_2020__358_1_45_0
ER  -

%0 Journal Article
%A Hafedh Khalfoun
%A Ismail Laraiedh
%T The linear $\protect \mathfrak{n}(1|N)$–invariant differential operators and $\protect \mathfrak{n}(1|N)$–relative cohomology
%J Comptes Rendus. Mathématique
%D 2020
%P 45-58
%V 358
%N 1
%I Académie des sciences, Paris
%R 10.5802/crmath.22
%G en
%F CRMATH_2020__358_1_45_0

Hafedh Khalfoun; Ismail Laraiedh. The linear $\protect \mathfrak{n}(1|N)$–invariant differential operators and $\protect \mathfrak{n}(1|N)$–relative cohomology. Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 45-58. doi : 10.5802/crmath.22. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.22/

Bibliographie
Cité par

[1] Boujemaa Agrebaoui; Nizar Ben Fraj; Salem Omri On the cohomology of the Lie superalgebra of contact vectorfields on $S^{1 | 2}$ , J. Nonlinear. Math. Phys., Volume 13 (2006) no. 1-4, pp. 523-534 | Zbl

[2] Imed Basdouri; Mabrouk Ben Ammar Cohomology of $𝔬𝔰𝔭 (1 | 2)$ with coefficients in $𝔇_{λ, μ}$ , Lett. Math. Phys., Volume 81 (2007) no. 3, pp. 239-251 | Zbl

[3] Imed Basdouri; Ismail Laraiedh; Othmen Ncib The linear $𝔞𝔣𝔣 (n | 1)$ -invariant differential operators on weighted densities on the superspace $ℝ^{1 | n}$ and $𝔞𝔣𝔣 (n | 1)$ -relative cohomology, Int. J. Geom. Methods Mod. Phys., Volume 10 (2013) no. 4, 1320004, 9 pages | Zbl

[4] Charles H. Conley Conformal symbols and the action of Contact vector fields over the superline, J. Reine Angew. Math., Volume 633 (2009), pp. 115-163 | MR | Zbl

[5] Dmitry B. Fuks Cohomology of infinite-dimensional Lie algebras, Plenum Press, 1986 | Zbl

[6] Hichem Gargoubi; Najla Mellouli; Valentin Ovsienko Differential operators on supercircle: conformally equivariant quantization and symbol calculus, Lett. Math. Phys., Volume 79 (2007) no. 1, pp. 51-65 | DOI | MR | Zbl

[7] Pavel Grozman; Dimitry Leites; Irina Shchepochkina Lie superalgebras of string theories, Acta. Math. Vietnam., Volume 26 (2001) no. 1, pp. 27-63 | MR | Zbl

[8] Pavel Grozman; Dimitry Leites; Irina Shchepochkina Invariant operators on supermanifolds and standard models, Multiple facets of quantization and supersymmetry, World Scientific, 2002, pp. 508-555 | DOI | Zbl

[9] Dimitry Leites Introduction to the theory of supermanifolds, Usp. Mat. Nauk, Volume 35 (1980) no. 1, pp. 3-57 translated in English in Russ. Math. Surv. 35 (1980), no. 1, p. 1-64 | MR | Zbl

[10] Valentin Ovsienko; Claude Roger Deforming the Lie algebra of vector fields on $S^{1}$ inside the Poisson algebra on $T^{*} S^{1}$ , Commun. Math. Phys., Volume 198 (1998) no. 1, pp. 97-110 | Zbl

[11] Valentin Ovsienko; Claude Roger Deforming the Lie algebra of vector fields on $S^{1}$ inside the Lie algebra of pseudodifferential operators on $S^{1}$ , Adv. Math. Sci., Volume 194 (1999), pp. 211-227 | Zbl

[12] Elena Poletaeva The analogs of Riemann and Penrose tensors on supermanifolds (2005) (https://arxiv.org/abs/math/0510165)

Cité par Sources :

Commentaires - Politique

Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !

Publier un nouveau commentaire:

Publier une nouvelle réponse: