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Poisson Dixmier-Moeglin equivalence from a topological point of view

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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. 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.

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Acknowledgments

Part of this research work was done during the first and second authors’ visit to Shanghai Center for Mathematical Sciences in June-July 2019. They are grateful for the invitations of the third author and wish to thank Fudan University for its hospitality. Juan Luo is supported by the NSFC (project 11901396). Xingting Wang is supported by Simons Collaboration Grant: 688403. Quanshui Wu is supported by the NSFC (project 11771085). The authors would also like to thank Jason Bell, Ken Brown, Ken Goodearl, and James Zhang for useful correspondence and comments on this paper. All the authors wish to thank the referee for valuable comments.

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Authors and Affiliations

  1. Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China

    Juan Luo

  2. Department of Mathematics, Howard University, 2400 Sixth St. NW, Washington DC, 20059, USA

    Xingting Wang

  3. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Quanshui Wu

Authors
  1. Juan Luo
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  2. Xingting Wang
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  3. Quanshui Wu
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Correspondence to Xingting Wang.

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Luo, J., Wang, X. & Wu, Q. Poisson Dixmier-Moeglin equivalence from a topological point of view. Isr. J. Math. 243, 103–139 (2021). https://doi.org/10.1007/s11856-021-2154-9

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  • DOI: https://doi.org/10.1007/s11856-021-2154-9

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