Poisson structures of multi-parameter symplectic and Euclidean spaces

TL;DR

This paper constructs and analyzes Poisson algebras as classical limits of multiparameter quantized spaces, describing their prime and symplectic ideals and showing their topological quotient structure.

Contribution

It introduces a class of Poisson algebras for multiparameter symplectic and Euclidean spaces and characterizes their ideal structures, linking quantum and classical geometries.

Findings

Prime Poisson ideals are classified.

Symplectic ideals are characterized.

Quantum spaces are topological quotients of classical spaces.

Abstract

A class of Poisson algebras considered as a Poisson version of the multiparameter quantized coordinate rings of symplectic and Euclidean $2n$-spaces is constructed and the prime Poisson ideals and the symplectic ideals of these Poisson algebras are described. As a result, it is shown that the multiparameter quantized symplectic and Euclidean $2n$-spaces are topological quotients of their classical spaces.