  Our aim in this thesis is to use the language of deformation-quantization to
understand certain quantized algebras by looking at properties of the
corresponding commutative ones, and conversely to obtain results about the
commutative algebras (upon which a Poisson structure is induced) using existing
results for the non-commutative ones. We consider two main cases: firstly, the
division ring of fractions of the quantum plane, which we view as a deformation
of the commutative field of rational functions in two variables with respect to
the bracket $\{x,y\} = xy$, and secondly, quantum matrices and their
semi-classical limits. In particular, we use the theory of $H$-stratification
to study the Poisson-prime and Poisson-primitive ideals of $\mathcal{O}(GL_3)$
and $\mathcal{O}(SL_3)$, and compare this to the corresponding results for
quantum matrices.
