Semiclassical limits of quantum affine spaces

Abstract

Semiclassical limits of generic multi-parameter quantized coordinate rings A=_q(kⁿ) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsion-free subgroup of k^×. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R=(kⁿ), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson-prime and Poisson-primitive spectra of R onto the prime and primitive spectra of~A. The Poisson-primitive spectrum of R is then identified with the space of symplectic cores in kⁿ in the sense of Brown and Gordon, and an example is presented (over ℂ) for which the Poisson-primitive spectrum of R is not homeomorphic to the space of symplectic leaves in kⁿ. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties.