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. Sánchez 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.

一个复杂的仿射泊松代数一个被说成满足泊松Dixmier-Moeglin等价若极大理想的泊松核一恰恰是在泊松主要频谱P.spec局部封闭的泊松素理想一个如果，而且，这些泊松素理想正是那些扩展的泊松中心恰好是复数的理想。

在本文中，我们提供了关于Poisson Dixmier-Moeglin等价物A的拓扑标准，该等价物以体素（P.spec A，⊆）以及其最大光谱上的辛叶或核心分层为依据。特别是，我们证明了泊松素数谱以及每个辛叶或核的Zariski拓扑可以检测任何复杂仿射泊松Poisson代数的泊松Dixmier-Moeglin等效性。此外，我们将复杂的仿射Poisson代数在[J. 贝尔，S。Launois，OLSánchez和B. Moosa，泊松代数，通过模型理论和微分代数几何，J。Eur。数学。Soc。（JEMS）19 （2017年，2019–2049年）到可交换微分代数的一般背景。