Date of Award

May 2013

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Allen D. Bell

Committee Members

Kevin McLeod, Ian Musson, Jeb Willenbring, Yi Ming Zou

Keywords

Artin-Schelter Regular, Deformation, Geometric Algebra, Homogenization, Matrix Congruence, Skew Polynomial Ring

Abstract

A central object in the study of noncommutative projective geometry is the (Artin-Schelter) regular algebra, which may be considered as a noncommutative version of a polynomial ring. We extend these ideas to algebras which are not necessarily graded. In particular, we define an algebra to be essentially regular of dimension d if its homogenization is regular of dimension d+1. Essentially regular algebras are described and it is shown that that they are equivalent to PBW deformations of regular algebras. In order to classify essentially regular algebras we introduce a modified version of matrix congruence, called sf-congruence, which is equivalent to affine maps between non-homogeneous quadratic polynomials. We then apply sf-congruence to classify homogenizations of 2-dimensional essentially regular algebras. We study the representation theory of essentially regular algebras and their homogenizations, as well as some peripheral algebras.

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