Comptes Rendus
Lie Algebras, Mathematical Physics
The linear 𝔫(1|N)–invariant differential operators and 𝔫(1|N)–relative cohomology
Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 45-58.

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 𝒮Ψ𝒟𝒪(S 1|N ) of superpseudodifferential operators with smooth coeffcients. We explicity give 1-cocycles spanning these cohomology spaces.

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 𝒮Ψ𝒟𝒪(S 1|N ) des opérateurs superpseudodifférentiels. Nous donnons explicitement les 1-cocycles engendrent ces espaces de cohomologie.

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DOI: 10.5802/crmath.22
Classification: 53D55, 14F10, 17B10, 17B68
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     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},
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     year = {2020},
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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/

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