On the Cohomology of the Lie Superalgebra of Contact Vector Fields on S1∣2S^{1|2}

We investigate the first cohomology space associated with the embedding of the Lie superalgebra \cK(2) of contact vector fields on the (1,2)-dimensional supercircle $S^{1\mid 2}$ in the Lie superalgebra \cS\Psi \cD \cO(S^{1\mid 2}) of superpseudodifferential operators with smooth coefficients. Following Ovsienko and Roger, we show that this space is ten-dimensional with only even cocycles and we give explicit expressions of the basis cocycles.Comment: Accepted for publication at the Journal of Nonlinear Mathematical Physic

The Binary Invariant Differential Operators on Weighted Densities on the superspace R1∣n\mathbb{R}^{1|n} and Cohomology

Over the $(1,n)$-dimensional real superspace, $n>1$, we classify $\mathcal{K}(n)$-invariant binary differential operators acting on the superspaces of weighted densities, where $\mathcal{K}(n)$ is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of %the Lie superalgebra $\mathcal{K}(n)$ with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities--a superisation of a result by Feigin and Fuchs. We explicitly give 1-cocycles spanning these cohomology spaces