Cohomology of $\mathfrak{osp}(1|2)$ acting on the space of bilinear differential operators on the superspace ${\mathbb{R}}^{1|1}$

Abstract

We compute the first cohomology of the ortosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ on the (1,1)-dimensional real superspace with coefficients in the superspace ${\mathfrak{D}}_{\lambda ,\nu ;\mu }$ of bilinear differential operators acting on weighted densities. This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on $\mathbb{R}{\mathbb{P}}^1$ acting on bilinear differential operators, International Journal of Geometric Methods in Modern Physics (2005), {\bf 2}; N 1, 23-40].