Twisted algebras of geometric algebras

TL;DR

This paper explores how twisted algebras of geometric algebras can be characterized by automorphisms of their point varieties, providing a classification for 3-dimensional geometric Artin-Schelter regular algebras.

Contribution

It establishes a link between twisted algebras and automorphisms of point varieties, enabling classification of certain geometric algebras.

Findings

Twisted algebra of a geometric algebra is determined by an automorphism of its point variety.

Classification of twisted 3-dimensional geometric Artin-Schelter regular algebras achieved.

Provides a method to construct non-algebraic twisting systems from generators and relations.

Abstract

A twisting system is one of the major tools to study graded algebras, however, it is often difficult to construct a (non-algebraic) twisting system if a graded algebra is given by generators and relations. In this paper, we show that a twisted algebra of a geometric algebra is determined by a certain automorphism of its point variety. As an application, we classify twisted algebras of $3$-dimensional geometric Artin-Schelter regular algebras up to graded algebra isomorphism.