Quantum and Poisson structures of multi-parameter symplectic and Euclidean spaces

A class of Poisson algebras A(n,Gamma)(P,Q) considered as a Poisson version of the multiparameter quantized coordinate rings K-n,Gamma(P,Q) of symplectic and Euclidean 2n-spaces is constructed and Poisson structures of A(n,Gamma)(P,Q) are described. In particular, it is proved that the prime Poisson and Poisson primitive spectra of A(n,Gamma)(P,Q) are homeomorphic to the prime and primitive spectra K-n,Gamma(P,Q) in the case when the multiplicative subgroup of k(x) generated by all parameters K-n,Gamma(P,Q) is torsion free and, as a corollary, that the prime and primitive spectra of K-n,Gamma(P,Q) are topological quotients of the prime and maximal spectra of the corresponding commutative polynomial ring. (C) 2008 Elsevier Inc. All rights reserved.