[Paper Review] Poisson Dixmier-Moeglin equivalence from a topological point of view

Juan Luo, Xingting Wang+1 more|arXiv (Cornell University)|Aug 19, 2019

Advanced Topics in Algebra28 references4 citations

TL;DR

This paper establishes a purely topological characterization of the Poisson Dixmier-Moeglin equivalence for complex affine Poisson algebras, showing that the Zariski topology on the Poisson prime spectrum and on symplectic leaves detects whether Poisson primitive, locally closed, and rational Poisson prime ideals coincide. The key result is a criterion based on the cardinality of minimal Poisson prime ideals over a given ideal, proving that the equivalence holds if and only if such ideals have finitely many minimal elements when their number is less than |ℂ|.

ABSTRACT

In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for $A$ in terms of the poset $({ m P. spec A}, \subseteq)$ 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), no. 7, 2019-2049] to the general context of a commutative differential algebra.

Motivation & Objective

• To provide a topological characterization of the Poisson Dixmier-Moeglin equivalence in complex affine Poisson algebras.
• To determine when Poisson primitive, locally closed, and rational Poisson prime ideals coincide using only topological properties of the Poisson prime spectrum.
• To generalize a weaker version of the equivalence from model-theoretic methods to commutative differential algebras.
• To show that the Zariski topology on the Poisson prime spectrum and on symplectic leaves fully detects the Poisson Dixmier-Moeglin equivalence.

Proposed method

• Introduce the notion of κ-separability for Zariski spaces and posets to handle cardinality conditions in the topology of the Poisson prime spectrum.
• Use the poset structure (P.spec A, ⊆) to analyze the relationships between Poisson prime ideals and their minimality.
• Analyze the symplectic leaf or core stratification on the maximal spectrum to relate geometric and topological properties.
• Apply results from commutative differential algebras to extend the Poisson Dixmier-Moeglin equivalence to a broader algebraic context.
• Use the fact that symplectic leaves are locally closed in the maximal spectrum to link topological closure with Poisson ideal properties.
• Leverage the Goldie quotient ring and Poisson center conditions to characterize rational Poisson prime ideals.

Experimental results

Research questions

• RQ1 Under what topological conditions on the Poisson prime spectrum does the Poisson Dixmier-Moeglin equivalence hold for a complex affine Poisson algebra?
• RQ2 Can the Zariski topology on the Poisson prime spectrum detect whether Poisson primitive ideals coincide with locally closed and rational Poisson prime ideals?
• RQ3 How does the cardinality of minimal Poisson prime ideals over a given Poisson prime ideal affect the validity of the Poisson Dixmier-Moeglin equivalence?
• RQ4 To what extent can the Poisson Dixmier-Moeglin equivalence be generalized from Poisson algebras to commutative differential algebras?
• RQ5 Do symplectic leaves or cores in the maximal spectrum provide sufficient topological information to detect the Poisson Dixmier-Moeglin equivalence?

Key findings

• The Poisson Dixmier-Moeglin equivalence holds for a complex affine Poisson algebra A if and only if every Poisson prime ideal with fewer than |ℂ| minimal Poisson prime ideals over it has only finitely many such minimal ideals.
• The Zariski topology on the Poisson prime spectrum (P.spec A) fully detects the Poisson Dixmier-Moeglin equivalence.
• The symplectic leaf or core stratification on the maximal spectrum also detects the Poisson Dixmier-Moeglin equivalence.
• The paper generalizes a weaker version of the Poisson Dixmier-Moeglin equivalence to the setting of commutative differential algebras.
• Examples such as the Sklyanin Poisson algebra and Poisson structures on elliptic curve quotients satisfy the equivalence, as their symplectic leaves are finite and algebraic.
• The center of a noncommutative algebra with a rational torus action satisfies the Poisson Dixmier-Moeglin equivalence when the action is finite and the center is finitely generated.