Ore Localizations and Irreducible Representations of the First Weyl Algebra

Abstract

<p>This thesis studies two problems for the first Weyl algebra A = A₁(C), namely, Ore localizations and irreducible representations.</p> <p>Our contribution to the first problem is that we find two collections of torsion theories which can be determined by Ore sets. The first consists of all torsion theories generated by classes of simple A-modules which contains either all C[q]-torsion or all C[p]-torsion simple A-modules, up to an automorphism of A (for instance, any torsion theory generated by all but countably many isomorphism classes of simple modules). The second consists of all torsion theories generated by classes of at most linear simple A-modules.</p> <p>The second part of the thesis studies the irreducible representations of A, i.e., the structure of simple A-modules. We generalize Block's result for linear simple modules, namely, that every linear simple module can be expressed in the form C[X,α⁻¹] for some α ∈ C[X], to arbitrary simple modules which satisfies two conditions which are necessary and sufficient. The second condition is stated in terms of two invariants of the similarity class corresponding to the given simple module, which are explicitly checkable. An important tool is an index theorem which relates two different realizations of the same simple module.</p>