Poisson brackets and Poisson spectra in polynomial algebras

TL;DR

This paper classifies all Poisson brackets on the polynomial algebra C[x,y,z] and characterizes their Poisson prime and primitive ideals for a significant class, supported by numerous examples.

Contribution

It provides a complete description of all Poisson brackets on C[x,y,z] and determines Poisson prime and primitive ideals for a broad class, advancing understanding of Poisson structures.

Findings

Complete classification of Poisson brackets on C[x,y,z]

Determination of Poisson prime and primitive ideals for key classes

Illustrative examples demonstrating the results

Abstract

Poisson brackets on the polynomial algebra C[x,y,z] are studied. A description of all such brackets is given and, for a significant class of Poisson brackets, the Poisson prime ideals and Poisson primitive ideals are determined. The results are illustrated by numerous examples.