Poisson algebras via model theory and differential-algebraic geometry

TL;DR

This paper investigates the Poisson Dixmier-Moeglin equivalence for complex affine Poisson algebras, revealing dimension-dependent behaviors and counterexamples, and employs differential-algebraic geometry and model theory techniques.

Contribution

It provides a complete answer to the equivalence question, constructs counterexamples in higher dimensions, and establishes a weaker version valid in all dimensions.

Findings

Poisson rational and primitive ideals coincide

Counterexamples exist in dimension four and higher

Full equivalence holds in dimension three or less

Abstract

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite $\operatorname{GK}$ dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.