  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.
