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. Vancliff's main objective is to add to the body of knowledge on regular algebras and to further the existing geometric techniques. In particular, she is interested in the graded-module category of such an algebra 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 is also interested in connections between this type of geometric analysis and that of various Poisson-geometric structures. Some of these activities entail the development of fundamental computational algorithms, and their implementation using a computer-algebra package, in order to enable explicit computation of linear schemes. 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. The solutions are often results that cannot be found by experimentation nor other methods, and often they are not numbers but are functions (e.g., differential operators or matrices), and so, in general, they do not commute. The business of seeking methods that find all solutions to any system of polynomial-style equations in non-commuting variables is called non-commutative algebra. The main idea to find the solutions is as follows. One associates to such a system of equations an abstract object called an algebra, which encodes all the properties of the system. To the algebra is associated abstract objects called modules, and these encode all the properties of the solutions. Hence, in order to find all the solutions, one should find and understand 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 some of the 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.