On the Dixmier-Moeglin equivalence for Poisson-Hopf algebras

We prove that the Poisson version of the Dixmier-Moeglin equivalence

holds for cocommutative a?ne Poisson-Hopf algebras. This is a ?rst step

towards understanding the symplectic foliation and the representation theory

of (cocommutative) a?ne Poisson-Hopf algebras. Our proof makes substantial

use of the model theory of ?elds equipped with ?nitely many possibly noncommuting

derivations. As an application, we show that the symmetric algebra of

a ?nite dimensional Lie algebra, equipped with its natural Poisson structure,

satis?es the Poisson Dixmier-Moeglin equivalence.