A Dixmier-moeglin Equivalence for Poisson Algebras with Torus Actions

A Poisson analog of the Dixmier-Moeglin equivalence is established for any affine Poisson algebra R on which an algebraic torus H acts rationally, by Poisson automorphisms, such that R has only finitely many prime Poisson H-stable ideals. In this setting, an additional characterization of the Poisson primitive ideals of R is obtained – they are precisely the prime Poisson ideals maximal in their H-strata (where two prime Poisson ideals are in the same H-stratum if the intersections of their H-orbits coincide). Further, the Zariski topology on the space of Poisson primitive ideals of R agrees with the quotient topology induced by the natural surjection from the maximal ideal space of R onto the Poisson primitive ideal space. These theorems apply to many Poisson algebras arising from quantum groups. The full structure of a Poisson algebra is not necessary for the results of this paper, which are developed in the setting of a commutative algebra equipped with a set of derivations.