Revista de la
Unión Matemática Argentina
On the second $\mathfrak{osp}(1|2)$-relative cohomology of the Lie superalgebra of contact vector fields on $\mathcal{C}^{1|1}$
Abderraouf Ghallabi, Nizar Ben Fraj, and Salem Faidi

Volume 68, no. 2 (2025), pp. 349–367    

Published online (final version): July 16, 2025

https://doi.org/10.33044/revuma.3888

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Abstract

Let $\mathcal{K}(1)$ be the Lie superalgebra of contact vector fields on the $(1,1)$-dimensional complex superspace; it contains the Möbius superalgebra $\mathfrak{osp}(1|2)$. We classify $\mathfrak{osp}(1|2)$-invariant superanti-symmetric binary differential operators from $\mathcal{K}(1)\wedge\mathcal{K}(1)$ to $\mathfrak{D}_{\lambda,\mu}$ vanishing on $\mathfrak{osp}(1|2)$, where $\mathfrak{D}_{\lambda,\mu}$ is the superspace of linear differential operators acting on the superspaces of weighted densities. This result allows us to compute the second differential $\mathfrak{osp}(1|2)$-relative cohomology of $\mathcal{K}(1)$ with coefficients in $\mathfrak{D}_{\lambda,\mu}$.

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