REGULARITY CRITERION AND CLASSIFICATION FOR ALGEBRAS OF JORDAN TYPE

We show that Artin–Schelter regularity of a $\mathbb{Z}$ -graded algebra can be examined by its associated $\mathbb{Z}$ r-graded algebra. We prove that there is exactly one class of four-dimensional Artin–Schelter regular algebras with two generators of degree one in the Jordan type. This class is strongly noetherian, Auslander regular, and Cohen–Macaulay. Their automorphisms and point modules are described.