𝔞𝔣𝔣(1|1)-Relative cohomology on ℝ1|1

Over the [Formula: see text]-dimensional real superspace [Formula: see text], we classify [Formula: see text]-invariant bilinear differential operators acting on the superspaces of weighted densities. We compute the second [Formula: see text]-relative cohomology space of [Formula: see text] with coefficients in the module of [Formula: see text]-densities [Formula: see text] on [Formula: see text], where [Formula: see text] is the Lie superalgebra of contact vector fields on [Formula: see text] and [Formula: see text] is the affine Lie superalgebra. This result allows us to compute the second [Formula: see text]-relative cohomology space of [Formula: see text] with coefficients in the Poisson superalgebra [Formula: see text]. We explicitly give 2-cocycles spanning these cohomology spaces.

THE MALCEV POISSON SUPERALGEBRA OF THE FREE MALCEV SUPERALGEBRA ON ONE ODD GENERATOR

A base of the Malcev Poisson superalgebra related with the free Malcev superalgebra on one odd generator is constructed. As a corollary, a pre-base, that is, a set of elements that spans the free alternative superalgebra on one odd generator is obtained.