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

Abstract

AbstractA class of Poisson algebras An,ΓP,Q considered as a Poisson version of the multiparameter quantized coordinate rings Kn,ΓP,Q of symplectic and Euclidean 2n-spaces is constructed and Poisson structures of An,ΓP,Q are described. In particular, it is proved that the prime Poisson and Poisson primitive spectra of An,ΓP,Q are homeomorphic to the prime and primitive spectra of Kn,ΓP,Q in the case when the multiplicative subgroup of k× generated by all parameters in Kn,ΓP,Q is torsion free and, as a corollary, that the prime and primitive spectra of Kn,ΓP,Q are topological quotients of the prime and maximal spectra of the corresponding commutative polynomial ring