A natural map from a quantized space onto its semiclassical limit

TL;DR

This paper constructs a deformation of a Poisson algebra and demonstrates that the natural map from the quantized space to its semiclassical limit can send prime ideals to non-Poisson prime ideals, revealing subtle differences between quantum and classical structures.

Contribution

It provides a specific example of a deformation where the natural map does not preserve Poisson primality, highlighting limitations in the correspondence between quantum and classical algebraic structures.

Findings

Constructed a deformation $B_q$ of a Poisson algebra $B_1$.

Found a prime ideal $P$ in $B_q$ with a non-Poisson prime image.

Showed the natural map can fail to preserve Poisson primality.

Abstract

Let $\widehat{\Gamma}$ be the natural map given in \cite[\S1]{Oh12}. Here, we construct a deformation $B_q$ of a Poisson algebra $B_1$ and a prime ideal $P$ of $B_q$ such that $\widehat{\Gamma}(P)$ is not a Poisson prime ideal of $B_1$.