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

Khalfoun, Hafedh; Laraiedh, Ismail

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]

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

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

Comptes Rendus. Mathématique, Tome 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 :

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

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     author = {Khalfoun, Hafedh and Laraiedh, Ismail},
     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},
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     year = {2020},
     doi = {10.5802/crmath.22},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/crmath.22/}
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Khalfoun, Hafedh; Laraiedh, Ismail. The linear $\protect \mathfrak{n}(1|N)$–invariant differential operators and $\protect \mathfrak{n}(1|N)$–relative cohomology. Comptes Rendus. Mathématique, Tome 358 (2020) no. 1, pp. 45-58. doi : 10.5802/crmath.22. http://www.numdam.org/articles/10.5802/crmath.22/

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