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Poisson algebras via model theory and differential-algebraic geometry

Bell, Jason, Launois, Stephane, Leon Sanchez, Omar, Moosa, Rahim (2017) Poisson algebras via model theory and differential-algebraic geometry. Journal of The European Mathematical Society, 19 . pp. 2019-2049. ISSN 1435-9855. E-ISSN 1435-9863. (doi:10.4171/JEMS/712) (KAR id:41364)


Brown and Gordon asked whether the Poisson Dixmier–Moeglin equivalence holds for any complex affine Poisson algebra, that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier–Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier–Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

Item Type: Article
DOI/Identification number: 10.4171/JEMS/712
Uncontrolled keywords: Poisson algebras, differential algebraic geometry, Dixmier–Moeglin equivalence, primitive ideal, model theory, Manin kernel
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QA Mathematics (inc Computing science) > QA564 Algebraic Geometry
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Stephane Launois
Date Deposited: 09 Jun 2014 15:34 UTC
Last Modified: 09 Dec 2022 06:25 UTC
Resource URI: (The current URI for this page, for reference purposes)

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