Semiclassical limits of quantized coordinate rings

TL;DR

This paper explores the relationship between quantized coordinate rings and their classical limits, proposing that primitive ideals correspond to symplectic cores rather than symplectic leaves, supported by various algebraic examples.

Contribution

It introduces the concept of symplectic cores as a replacement for symplectic leaves in the study of primitive spectra of quantized coordinate rings.

Findings

Primitive spectrum of A is homeomorphic to symplectic cores in V

Symplectic cores better capture algebraic structures than symplectic leaves

Examples support the conjecture relating primitive ideals and symplectic cores

Abstract

This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety V consists of the classical coordinate ring O(V) equipped with an associated Poisson structure. Conjectured relationships between primitive ideals of a generic quantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit Poisson structure on O(V)) are discussed, as are breakdowns in the connections when the symplectic leaves are not algebraic. This prompts replacement of the differential-geometric concept of symplectic leaves with the algebraic concept of symplectic cores, and a reformulated conjecture is proposed: The primitive spectrum of A should be homeomorphic to the space of symplectic cores in V, and to the Poisson-primitive spectrum of O(V). Various examples, including both quantized coordinate rings and enveloping algebras of solvable Lie algebras, are analyzed to support the choice of symplectic cores to replace symplectic leaves.