This research is concerned with the study of new classes of graded algebras including Sklyanin algebras, regular algebras and algebras related to quantum groups. The goal is to understand their representation theory in geometric terms. The principal investigator will study the finite dimensional simple modules of Sklyanin algebras. He will examine higher dimensional Sklyanin algebras for properties such as regularity and noetherian chain conditions. He will also examine their properties from a deformation perspective. 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.
