# On the Dixmier-Moeglin equivalence for Poisson-Hopf algebra

## TL;DR

This paper establishes the Poisson Dixmier-Moeglin equivalence for cocommutative affine Poisson-Hopf algebras, advancing understanding of their symplectic foliation and representation theory using model theory techniques.

## Contribution

It proves the equivalence for a broad class of Poisson-Hopf algebras and applies the result to symmetric algebras of finite dimensional Lie algebras.

## Key findings

- Poisson Dixmier-Moeglin equivalence holds for cocommutative affine Poisson-Hopf algebras
- Symmetric algebra of a finite dimensional Lie algebra satisfies the equivalence
- Model theory of fields with derivations is used in the proof

## Abstract

We prove that the Poisson version of the Dixmier-Moeglin equivalence holds for cocommutative affine Poisson-Hopf algebras. This is a first step towards understanding the symplectic foliation and the representation theory of (cocommutative) affine Poisson-Hopf algebras. Our proof makes substantial use of the model theory of fields equipped with finitely many possibly noncommuting derivations. As an application, we show that the symmetric algebra of a finite dimensional Lie algebra, equipped with its natural Poisson structure, satisfies the Poisson Dixmier-Moeglin equivalence.