Weighted Poisson polynomial rings

TL;DR

This paper classifies Poisson structures on weighted polynomial rings defined by homogeneous potentials, analyzes their rigidity, automorphisms, and computes Poisson cohomology for new classes, advancing understanding of Poisson algebra structures.

Contribution

It provides a comprehensive classification of potentials with isolated singularities, studies rigidity and automorphisms, and computes Poisson cohomology for new classes of weighted polynomial rings.

Findings

Classified potentials of specific degrees with isolated singularities.

Determined Poisson automorphism groups for certain potentials.

Computed Poisson cohomology groups for new classes of Poisson polynomial algebras.

Abstract

We discuss Poisson structures on a weighted polynomial algebra $A:=\Bbbk[x, y, z]$ defined by a homogeneous element $\Omega\in A$, called a potential. We start with classifying potentials $\Omega$ of degree deg$(x)+$deg$(y)+$deg$(z)$ with any positive weight (deg$(x)$, deg$(y)$, deg$(z)$) and list all with isolated singularity. Based on the classification, we study the rigidity of $A$ in terms of graded twistings and classify Poisson fraction fields of $A/(\Omega)$ for irreducible potentials. Using Poisson valuations, we characterize the Poisson automorphism group of $A$ when $\Omega$ has an isolated singularity extending a nice result of Makar-Limanov-Turusbekova-Umirbaev. Finally, Poisson cohomology groups are computed for new classes of Poisson polynomial algebras.