First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators

M.  Boujelben

doi:10.37863/umzh.v74i6.6052

M. Boujelben Univ. de Sfax, Tunisia https://orcid.org/0000-0002-5244-4477

DOI: https://doi.org/10.37863/umzh.v74i6.6052

Keywords: Cohomology, Orthosymplectic superalgebra, superpseudodifferential operators, Poisson superalgebra

Abstract

UDC 515.12
We investigate the first cohomology space associated with the embedding of the Lie orthosymplectic superalgebra $\mathfrak{osp}(n|2)$ on the $(1,n)$-dimensional superspace $\mathbb{R}^{1|n}$ in the Lie superalgebra $ \mathcal{S}\Psi\mathcal{DO}(n)$ (for $n \geq 4$ ) of superpseudodifferential operators with smooth coefficients.
Following Ovsienko and Roger, we give explicit expressions of the basis cocycles.
This work is the simplest generalization of a result by Basdouri [First space cohomology of the orthosymplectic Lie superalgebra in the Lie superalgebra of superpseudodifferential operators, Algebras and Representation Theory, 16, 35-50 (2013)].

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