PBW Deformations of Artin-Schelter Regular Algebras and Their Homogenizations

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.