Supertransvectants, cohomology, and deformations

Abstract

Over the (1, N)-dimensional real superspace, N = 2, 3, we classify \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(N|2)$\end{document}osp(N|2)-invariant binary differential operators acting on the superspaces of weighted densities, where \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(N|2)$\end{document}osp(N|2) is the orthosymplectic Lie superalgebra. This result allows us to compute the first differential \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(N|2)$\end{document}osp(N|2)-relative cohomology of the Lie superalgebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}(N)$\end{document}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 \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(3|2)$\end{document}osp(3|2)-trivial deformations of the \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}(3)$\end{document}K(3)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(3|2)$\end{document}osp(3|2)-trivial deformation of this \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}(3)$\end{document}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 \documentclass[12pt]{minimal}\begin{document}$\mathbb {K}^{1|1}$\end{document}K1|1 and deformations of the superspace of symbols,” J. Nonlinear Math. Phys. 16, 373 (2009)10.1142/S1402925109000431].