Geodesic
Poisson algebras via model theory and differential-algebraic geometry
Journal of the European Mathematical Society, Tome 19 (2017) no. 7, pp. 2019-2049.

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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.
DOI : 10.4171/jems/712
Classification : 17-XX, 03-XX, 12-XX, 14-XX
Keywords: Poisson algebras, differential algebraic geometry, Dixmier–Moeglin equivalence, primitive ideal, model theory, Manin kernel 
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Jason Bell; Stéphane Launois; Omar León Sánchez; Rahim Moosa. Poisson algebras via model theory and differential-algebraic geometry. Journal of the European Mathematical Society, Tome 19 (2017) no. 7, pp. 2019-2049. doi : 10.4171/jems/712. http://geodesic.mathdoc.fr/articles/10.4171/jems/712/

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