Poisson spectra in polynomial algebras

TL;DR

This paper investigates a broad class of Poisson brackets on polynomial algebras, characterizes their prime and primitive ideals, and proves they satisfy the Poisson Dixmier-Moeglin equivalence, advancing understanding of their algebraic structure.

Contribution

It identifies and analyzes a significant class of Poisson brackets on polynomial algebras, determining their prime and primitive ideals and establishing the Poisson Dixmier-Moeglin equivalence.

Findings

Poisson prime and primitive ideals are explicitly characterized.

The studied Poisson algebras satisfy the Poisson Dixmier-Moeglin equivalence.

Provides structural insights into polynomial Poisson algebras.

Abstract

A significant class of Poisson brackets on the polynomial algebra $\C[x_1,x_2,..., x_n]$ is studied and, for this class of Poisson brackets, the Poisson prime ideals and Poisson primitive ideals are determined. Moreover it is established that these Poisson algebras satisfy the Poisson Dixmier-Moeglin equivalence.