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

TL;DR

This paper investigates the deformations of the Lie superalgebra of contact vector fields on a (1,1)-dimensional superspace, calculating obstructions, the miniversal deformation algebra, and the first differential cohomology space.

Contribution

It provides a complete analysis of deformations of the $\mathcal{K}(1)$-modules of symbols, including obstructions, the miniversal deformation, and cohomology computations.

Findings

Calculated obstructions for integrability of deformations.

Determined the algebra of the miniversal deformation.

Computed the first differential cohomology space $H^1_{diff}$.

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)