Artin-Schelter regular algebras of dimension five with two generators

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 Gröbner basis computations. All the algebras we obtained are strongly noetherian, Auslander regular, and Cohen-Macaulay. One of the results provides an answer to Fløystad-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.