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Our aim in this thesis is to use the language of deformation-quantization tounderstand certain quantized algebras by looking at properties of thecorresponding commutative ones, and conversely to obtain results about thecommutative algebras (upon which a Poisson structure is induced) using existingresults for the non-commutative ones. We consider two main cases: firstly, thedivision ring of fractions of the quantum plane, which we view as a deformationof the commutative field of rational functions in two variables with respect tothe bracket $\{x,y\} = xy$, and secondly, quantum matrices and theirsemi-classical limits. In particular, we use the theory of $H$-stratificationto 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 forquantum matrices.

The $q$-Division Ring, Quantum Matrices and Semi-classical Limits