Poisson Dixmier-Moeglin equivalence from a topological point of view

Abstract

A complex affine Poisson algebra A is said to satisfy the Poisson Dixmier-Moeglin equivalence if the Poisson cores of maximal ideals of A are precisely those Poisson prime ideals that are locally closed in the Poisson prime spectrum P.spec A and if, moreover, these Poisson prime ideals are precisely those whose extended Poisson centers are exactly the complex numbers. In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for A in terms of the poset (P.spec A, ⊆) and the symplectic leaf or core stratification on its maximal spectrum. In particular, we prove that the Zariski topology of the Poisson prime spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover, we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for a complex affine Poisson algebra proved in [J. Bell, S. Launois, O. L. Sanchez and B. Moosa, Poisson algebras via model theory and differential-algebraic geometry, J. Eur. Math. Soc. (JEMS) 19 (2017), 2019–2049] to the general context of a commutative differential algebra.