Artin-Schelter regular algebras of dimension five with two generators

TL;DR

This paper classifies five-dimensional Artin-Schelter regular algebras with two generators using Hilbert-driven Gr"obner basis methods, revealing their algebraic properties and connecting Lyndon words to these structures.

Contribution

It provides a complete classification of such algebras under a specific grading, answering a known question and linking Lyndon words to Artin-Schelter regularity.

Findings

All classified algebras are strongly noetherian, Auslander regular, and Cohen-Macaulay.

The classification addresses Fl{\

Abstract

We study and classify Artin-Schelter regular algebras of dimension five with two generators under an additional $\mathbb Z^2$-grading by Hilbert driven Gr\"{o}bner basis computations. All the algebras we obtained are strongly noetherian, Auslander regular, and Cohen-Macaulay. One of the results provides an answer to Fl{\o}ystad-Vatne's question in the context of $\mathbb Z^2$-grading. Our results also achieve a connection between Lyndon words and Artin-Schelter regular algebras.