Poisson (co)homology of polynomial Poisson algebras in dimension four : Sklyanin's case

TL;DR

This paper computes the Poisson (co)homology for a specific polynomial Poisson structure in four dimensions, focusing on Sklyanin's case with a complete intersection and isolated singularity.

Contribution

It provides the first detailed computation of Poisson (co)homology for Sklyanin's polynomial Poisson algebra in four dimensions.

Findings

Explicit Poisson (co)homology groups obtained

Identification of singularity's impact on cohomology

New methods for computing Poisson (co)homology in complex cases

Abstract

In this paper, we compute the Poisson (co)homology of a polynomial Poisson structure given by two Casimir polynomial functions which define a complete intersection with an isolated singularity.