An application of the model theory of differential fields to Poisson algebras

An affine complex Poisson algebra is a finitely generated commutative integral complex algebra equipped with a Lie bracket which is a derivation in each argument. Motivated by work of Dixmier and Moeglin on noncommutative algebras in the seventies, Gordon and Brown asked in 2003 about the equivalence of three properties of prime Poisson ideals: (1) the topological property of being locally closed in the Poisson spectrum, (2) the representation-theoretic property of "primitivity" , and (3) the algebraic property of "rationality". It was already known that (1) implies (2) implies (3). Using families of Manin Kernels in differentially closed fields we show that there are in general rational Poisson prime ideals that are not locally closed. On the positive side, using a version of Hrushovski's finiteness theorem on codegree one differential subvarieties (the unpublished "Jouanolou's theorem" manuscript), we prove that the equivalence does hold after a natural weakening of  (1). I will report on these results, which are joint work with Jason Bell, Stephane Launois, and Omar Leon Sanchez.

 

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