The Dixmier-Moeglin equivalence and a Gel'fand-Kirillov problem for Poisson polynomial algebras

The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, including semiclassical limits of quantum matrices, quantum symplectic and euclidean spaces, quantum symmetric and antisymmetric matrices. For a similarly large class of Poisson polynomial rings, it is proved that the quotient field of the algebra (respectively, of any Poisson prime factor ring) is a rational function field F(x(1), ..., x(n)) over the base field (respectively, over an extension field of the base field) with {x(i),x(j)} = lambda(ij)x(i)x(j)for suitable scalars lambda(ij), thus establishing a quadratic Poisson version of the Gel'fand-Kirillov problem. Finally, partial solutions to the isomorphism problem for Poisson fields of the type just mentioned are obtained.