Let k be a field of characteristic zero and q ∈ kx not a root of unity. We may obtain non-commutative 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 non-commutative 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 deformation-quantization 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 non-commutative ones. The q-division ring kq(x; y) is of particular interest to us, being one of the easiest infinite-dimensional 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 q-division 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 k-algebra 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 Poisson-Noether 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 non-bijective conjugation maps. We also consider Poisson-prime and Poisson-primitive ideals of the coordinate rings O(GL3) and O(SL3), where the Poisson bracket is induced from the non-commutative 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 Poisson-primitive (resp. Poisson-prime) ideals of their semi-classical limits. We prove that there is a natural bijection from the Poisson-primitive 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:618048 |
| Date | January 2014 |
| Creators | Fryer, Sian |
| Contributors | Premet, Alexander; Stafford, Toby |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.research.manchester.ac.uk/portal/en/theses/the-qdivision-ring-quantum-matrices-and-semiclassical-limits(2306256f-4fbc-44c8-9cad-10377cf159a2).html |
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