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Over the -dimensional real superspace, we compute the cohomology space of the affine Lie superalgebra with coefficient in a large class of -modules . We apply our results to the module of weight densities and the module of linear differential operators acting on a superspace of weighted densities. We study nontrivial deformations of the natural action of the Lie superalgebra on the direct sum of the superspaces of weighted densities. [ABSTRACT FROM AUTHOR]

Over the (1, N)-dimensional real superspace, $${N \geqq 3}$$ , we study non-trivial deformations of the natural action of the orthosymplectic Lie superalgebra $${\mathfrak{osp}(N|2)}$$ on the direct sum of the superspaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action and prove that any formal deformation is equivalent to its infinitisemal part. Likewise we study the same problem for the Lie superalgebra $${\mathcal{K}(N)}$$ of contact vector fields instead of $${\mathfrak{osp}(N|2)}$$ getting the same results. This work is the simplest generalization of a result by I. Basdouri and M. Ben Ammar [] and F. Ammar and K. Kammoun []. [ABSTRACT FROM AUTHOR]