Over the (1, N)-dimensional real superspace, N = 2, 3, we classify

$\mathfrak {osp}(N|2)$
osp(N|2)-invariant binary differential operators acting on the superspaces of weighted densities, where
$\mathfrak {osp}(N|2)$
osp(N|2) is the orthosymplectic Lie superalgebra. This result allows us to compute the first differential
$\mathfrak {osp}(N|2)$
osp(N|2)-relative cohomology of the Lie superalgebra
$\mathcal {K}(N)$
K(N) of contact vector fields with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We classify generic formal
$\mathfrak {osp}(3|2)$
osp(3|2)-trivial deformations of the
$\mathcal {K}(3)$
K(3)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal
$\mathfrak {osp}(3|2)$
osp(3|2)-trivial deformation of this
$\mathcal {K}(3)$
K(3)-module is equivalent to its infinitesimal part. This work is the simplest generalization of a result by the first author et al. [Basdouri, I., Ben Ammar, M., Ben Fraj, N., Boujelbene, M., and Kammoun, K., “Cohomology of the Lie superalgebra of contact vector fields on
$\mathbb {K}^{1|1}$
K1|1 and deformations of the superspace of symbols,” J. Nonlinear Math. Phys. 16, 373 (2009) https://doi.org/10.1142/S1402925109000431].

Topics
Associative algebra, Lie algebras, Algebraic topology, General topology, Operator theory, Vector fields, Algebraic geometry, Functions and mappings, Particle symmetry, Maurer-Cartan equation
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2013
American Institute of Physics
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