Deformations of Poisson algebras

TL;DR

This paper classifies $Z_2$-graded Poisson structures on polynomial algebras, explores their cohomology, deformations, and $P_$-algebra structures, highlighting differences from the classical ungraded case.

Contribution

It provides classifications, cohomology computations, and deformation analyses of $Z_2$-graded Poisson structures, extending understanding in graded algebra contexts.

Findings

Classified small-dimensional $Z_2$-graded Poisson structures.

Computed associated Poisson cohomology groups.

Identified deformation and $P_$-algebra structures in specific cases.

Abstract

We study $\mathbb Z_2$-graded Poisson structures defined on $\mathbb Z_2$-graded commutative polynomial algebras. In small dimensional cases, we exhibit classifications of such Poisson structures, obtain the associated Poisson $\mathbb Z_2$-graded cohomology and in some cases, deformations of these Poisson brackets and $P_\infty$-algebra structures. We highlight differences and analogies between this $\mathbb Z_2$-graded context and the non graded context, by studying for example the links between Poisson cohomology and singularities.