Deforming the Lie Superalgebra $\mathcal{K}(1)$-Modules Of Symbols

Abstract

We study non-trivial deformations of the natural action of the Lie superalgebra $mathcalK(1)$ of contact vector fields on the (1,1)-dimensional superspace $mathbbR^{1|1}$ of th espace of symbols. We calculate obstructions for integrability of infinitesimal multi-parameter deformation and determine the complete commutative algebra corresponding to the miniversal deformation in the sense of A. Fialowski. Besides, we compute the first even differential cohomology space $mathrmH^1_{mathrm{diff}}(cK(1);widetilde{cD}_{lambda,mu})$ of the Lie superalgebra $mathcalK(1)$ of contact vector fields on the (1,1)-dimensional superspace $mathbbR^{1|1}$ with coefficients in the superspace $widetilde{mathcalD}_{lambda,mu}$ of linear differential operators from the superspace of weighted densities $\fF_{\lamda}$ to $\fF_{\mu}$. (To appear in Journal of Generalized Lie Theory and Applications)