Mathematical Sciences: Finite Dimensional Simple Modules andPrimitive Ideals

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This research is concerned with the study of new classes of noncommutative noetherian rings. An underlying goal is to show that there are many algebras which have a structure similar to the enveloping algebra of a semisimple Lie algebra. Among those to be studied are some having primitive factor rings which have an associated graded algebra which is the coordinate ring of a Kleinian singularity. In addition, the enveloping algebras of Lie superalgebras, and q-analogues of enveloping algebras will be studied with the long-term goal of classifying the space of primitive ideals. This research is in the general area of noncommutative ring theory. The rings considered in the project are of interest in many areas of mathematics including algebraic geometry. Given a curve, one of these rings can be associated with certain points on the curve. A better understanding of these rings and this association will be useful in determining the geometry of a given curve.