Cohomologies of the Poisson superalgebra on (2,n)-superdimensional spaces

TL;DR

This paper investigates the cohomology spaces of the Poisson superalgebra on (2,n)-superdimensional spaces, focusing on the zeroth, first, and second cohomology in the adjoint representation for nondegenerate brackets.

Contribution

It provides explicit calculations of the low-degree cohomology spaces of the Poisson superalgebra on Grassmann-valued functions with compact support.

Findings

Cohomology spaces are explicitly computed for the case of nondegenerate constant Poisson superbrackets.

The zeroth, first, and second cohomology spaces are characterized in the adjoint representation.

Results contribute to understanding the algebraic structure and deformation theory of Poisson superalgebras.

Abstract

Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on R^2 are investigated under suitable continuity restrictions on cochains. The zeroth, first, and second cohomology spaces in the adjoint representation of the Poisson superalgebra are found for the case of a nondegenerate constant Poisson superbracket.