Title:

The qdivision ring, quantum matrices and semiclassical limits

Let k be a field of characteristic zero and q ∈ kx not a root of unity. We may obtain noncommutative counterparts of various commutative algebras by twisting the multiplication using the scalar q: one example of this is the quantum plane kq[x; y], which can be viewed informally as the set of polynomials in two variables subject to the relation xy = qyx. We may also consider the full localization of kq[x; y], which we denote by kq(x; y) or D and view as the noncommutative analogue of k(x; y), and also the quantization Oq(Mn) of the coordinate ring of n x n matrices over k. Our aim in this thesis will be to use the language of deformationquantization to understand the quantized algebras by looking at certain properties of the commutative ones, and conversely to obtain results about the commutative algebras (upon which a Poisson structure is induced) using existing results for the noncommutative ones. The qdivision ring kq(x; y) is of particular interest to us, being one of the easiest infinitedimensional division rings to define over k. Very little is known about such rings: in particular, it is not known whether its fixed ring under a finite group of automorphisms should always be isomorphic to another qdivision ring (possibly for a different value of q) nor whether the left and right indexes of a subring E ? D should always coincide. We define an action of SL2(Z) by kalgebra automorphisms on D and show that the fixed ring of D under any finite group of such automorphisms is isomorphic to D. We also show that D is a deformation of the commutative field k(x; y) with respect to the Poisson bracket fy; xg = yx and that for any finite subgroup G of SL2(Z) the xed ring DG is in turn a deformation of k(x; y)G. Finally, we describe the Poisson structure of the fixed rings k(x; y)G, thus answering the PoissonNoether question in this case. A number of interesting results can be obtained as a consequence of this: in particular, we are able to answer several open questions posed by Artamonov and Cohn concerning the structure of the automorphism group Aut(D). They ask whether it is possible to define a conjugation automorphism by an element z 2 LnD, where L is a certain overring of D, and whether D admits any endomorphisms which are not bijective. We answer both questions in the affirmative, and show that up to a change of variables these endomorphisms can be represented as nonbijective conjugation maps. We also consider Poissonprime and Poissonprimitive ideals of the coordinate rings O(GL3) and O(SL3), where the Poisson bracket is induced from the noncommutative multiplication on Oq(GL3) and Oq(SL3) via deformation theory. This relates to one case of a conjecture made by Goodearl, who predicted that there should be a homeomorphism between the primitive (resp. prime) ideals of certain quantum algebras and the Poissonprimitive (resp. Poissonprime) ideals of their semiclassical limits. We prove that there is a natural bijection from the Poissonprimitive ideals of these rings to the primitive ideals of Oq(GL3) and Oq(SL3), thus laying the groundwork for verifying this conjecture in these cases.
