Cohomology of the Lie Superalgebra of Contact Vector Fields on 𝕂^1|1 and Deformations of the Superspace of Symbols

Abstract

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional real or complex superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but osp(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal osp(1|2)-trivial deformations of the K(1)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal osp(1|2)-trivial deformation of this K(1)-module is equivalent to a polynomial one of degree ≤ 4. This work is the simplest superization of a result by Bouarroudj [On sl (2)

osp(1|2)-trivial deformation of osp(2Almoneef, Abdaoui, Ghallabi 2024Heliyon0000View full textAdd to dashboardCite

“…In 2009, Basdouri and al. [6] classified the deformations of -modules of symbols trivial on . They are proved that the conditions of integrability of the infinitesimal deformation trivial on are necessary and sufficient.…”

Second Cohomology Space of the Lie Superalgebra of contact vector fields $\mathcal{K}(2)$ with coefficients in the superspace of weighted densities on $S^{1|2}$

“…Therefore, the spaces of weighted densities F n λ are also K(n − 1)-modules. In [3,5] it was proved that, as K(1)-module, we have…”