Semiclassical Limits of Quantum Affine Spaces

TL;DR

This paper explores the semiclassical limits of quantum affine spaces, establishing connections between Poisson structures and prime spectra, and extends these results to quantum affine toric varieties.

Contribution

It constructs and analyzes semiclassical limits of quantum affine spaces and toric varieties, linking Poisson spectra to prime spectra and identifying symplectic cores.

Findings

Homeomorphisms between Poisson and prime spectra are established.

Poisson primitive spectrum corresponds to symplectic cores in affine space.

Counterexample shows differences between Poisson primitive spectrum and symplectic leaves.

Abstract

Semiclassical limits of generic multiparameter quantized coordinate rings A = O_q(k^n) 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 torsionfree subgroup of k*. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R = O(k^n), 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 k^n in the sense of Brown and Gordon, and an example is presented (over the complex numbers) for which the Poisson primitive spectrum of R is not homeomorphic to the space of symplectic leaves in k^n. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties.