# 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)

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## Abstract

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 10.4171/JEMS/712 Poisson algebras, differential algebraic geometry, Dixmier–Moeglin equivalence, primitive ideal, model theory, Manin kernel Q Science > QA Mathematics (inc Computing science) > QA150 AlgebraQ Science > QA Mathematics (inc Computing science) > QA564 Algebraic Geometry Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Pure Mathematics Stephane Launois 09 Jun 2014 15:34 UTC 15 Jan 2020 14:34 UTC https://kar.kent.ac.uk/id/eprint/41364 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-7252-8515