 
Summary: THE DIXMIERMOEGLIN EQUIVALENCE
AND A GEL'FANDKIRILLOV PROBLEM
FOR POISSON POLYNOMIAL ALGEBRAS
K. R. Goodearl and S. Launois
Abstract. 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 firstnamed author, this estab
lishes the Poisson DixmierMoeglin 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 poly
nomial rings, it is proved that the quotient field of the algebra (respectively, of any Poisson
prime factor ring) is a rational function field F(x1, . . . , xn) over the base field (respectively,
over an extension field of the base field) with {xi, xj} = ijxixj for suitable scalars ij,
thus establishing a quadratic Poisson version of the Gel'fandKirillov problem. Finally, par
tial 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 klinear.
