Regular Algebras

University Of Texas At Arlington, Arlington TX

Investigators

Abstract

This award supports the research of Michaela Vancliff to work in non-commutative algebra, with special emphasis on problems arising from the theory of regular algebras and non-commutative algebraic geometry. She is interested in the graded-module category viewed as a geometric space, with certain graded modules playing the role of geometric objects. The linear geometric modules (point modules, line modules, etc) are parametrized by so-called linear schemes. Vancliff plans to study how the structure and role of higher-dimensional linear schemes generalize the structure and role of point schemes. She intends to produce algebro-geometric techniques that allow the easy construction of regular algebras of global dimension four that have finitely many points and a one-parameter family of line modules; such techniques would allow researchers in the field to easily create examples on which to test their conjectures. An underlying theme of her research is to classify the line schemes that arise for "generic" quadratic regular algebras of global dimension four. Vancliff is also interested in connections between this type of geometry and that of various Poisson-geometric structures. Systems of polynomial-style equations and their solutions play a critical role in almost every scientific field, such as statistical mechanics, elementary-particle physics, quantum mechanics, robotics, crystallography, networking, etc. Often, the solutions cannot be found by experimentation, and often they are not numbers but are functions (e.g., differential operators or matrices), and so, in general, they do not commute. The science of seeking methods that find all solutions to any system of polynomial-style equations in non-commuting variables is called non-commutative algebra. To find the solutions, the main idea is as follows. One associates to such a system of equations a certain algebra; one that encodes all the properties of the original equations. Associated to this algebra are modules, and these encode all the properties of the original solutions. Hence, in order to find all the solutions, one should find all the modules for the associated algebra. In many of the applications, the algebras that arise in this way tend to share certain properties; they are called regular algebras and are the main focus of Vancliff's projects. One of the goals of non-commutative algebraic geometry, the subfield in which Vancliff works, is to use geometric techniques to find certain modules (point modules, line modules, etc) of the regular algebra, and then to use those modules to find the modules giving the solutions to the original system of equations. Vancliff's underlying goal is to improve on these geometric techniques and to understand better how they relate to the structure of the category of modules.