Regularity criterion and classification for algebras of Jordan type

TL;DR

This paper investigates the regularity and classification of certain graded algebras, revealing a unique class of four-dimensional Artin-Schelter regular algebras with specific properties in the Jordan case.

Contribution

It introduces a regularity criterion linking $Z$-graded and $Z^r$-graded algebras and classifies a unique class of four-dimensional Artin-Schelter regular Jordan algebras.

Findings

Exactly one class of four-dimensional Artin-Schelter regular Jordan algebras with two degree-one generators.

The class is strongly noetherian, Auslander regular, and Cohen-Macaulay.

Automorphisms and point modules of these algebras are described.

Abstract

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 case. This class is strongly noetherian, Auslander regular, and Cohen-Macaulay. Their automorphisms and point modules are described.