2 4 M ay 2 00 7 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 semiclas-sical 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, such as 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 } = λ ij x i x j for suitable scalars λ 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. 0. Introduction Fix a base field k of characteristic zero throughout. All algebras are assumed to be over k, and all relevant maps (automorphisms, derivations, etc.) are assumed to be k-linear. Recall that a Poisson algebra (over k) is a commutative k-algebra A equipped with a Lie bracket {−, −} which is a derivation (for the associative multiplication) in each variable. (see §1.1 for more detail on the conditions satisfied by such a bracket). Many such Poisson algebras are semiclassical limits of quantum algebras, and these provide