Bell, Jason, Launois, Stephane, Leon Sanchez, Omar, Moosa, Rahim (2017) Poisson algebras via model theory and differentialalgebraic geometry. Journal of The European Mathematical Society, 19 . pp. 20192049. ISSN 14359855. EISSN 14359863. (doi:10.4171/JEMS/712) (KAR id:41364)
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Official URL http://dx.doi.org/10.4171/JEMS/712 
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 differentialalgebraic 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:  Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Pure Mathematics 
Depositing User:  Stephane Launois 
Date Deposited:  09 Jun 2014 15:34 UTC 
Last Modified:  15 Jan 2020 14:34 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/41364 (The current URI for this page, for reference purposes) 
Launois, Stephane:  https://orcid.org/0000000172528515 
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