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

TL;DR

This paper calculates the first cohomology of the superalgebra rak{osp}(1|2) acting on bilinear differential operators on a 1|1-dimensional superspace, extending classical results to a supergeometric context.

Contribution

It provides the first cohomology computation for rak{osp}(1|2) on bilinear differential operators, generalizing known results from vector fields to superalgebras.

Findings

First cohomology groups explicitly computed

Extension of classical cohomology results to superalgebra context

Provides foundational results for supergeometry and representation theory

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 $\frak{D}_{\lambda,\nu;\mu}$ 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].