Graded Clifford Algebras and Graded Skew Clifford Algebras and Their Role in the Classification of Artin-Schelter Regular Algebras

TL;DR

This survey explores graded Clifford and skew Clifford algebras, their properties related to Artin-Schelter regularity, and their role in classifying noncommutative regular algebras, highlighting key connections to point modules and quadrics.

Contribution

It provides a comprehensive overview of conditions for these algebras to be Artin-Schelter regular and links point modules to noncommutative quadrics, extending previous foundational results.

Findings

Characterization of Artin-Schelter regularity conditions

Construction methods for GCAs with finitely many point modules

Generalization of point module classification to skew Clifford algebras

Abstract

This paper is a survey of work done on $\mathbb{N}$-graded Clifford algebras (GCAs) and $\mathbb{N}$-graded \textit{skew} Clifford algebras (GSCAs) \cite{VVW, SV, CaV, NVZ, VVe1, VVe2}. In particular, we discuss the hypotheses necessary for these algebras to be Artin Schelter-regular \cite{AS, ATV1} and show how certain `points' called, point modules, can be associated to them. We may view an AS-regular algebra as a noncommutative analog of the polynomial ring. We begin our survey with a fundamental result in \cite{VVW} that is essential to subsequent results discussed here: the connection between point modules and rank-two quadrics. Using, in part, this connection the authors in \cite{SV} provide a method to construct GCAs with finitely many distinct isomorphism classes of point modules. In \cite{CaV}, Cassidy and Vancliff introduce a quantized analog of a GCA, called a graded \textit{skew} Clifford algebra and Nafari et al. \cite{NVZ} show that most Artin Schelter-regular algebras of global dimension three are either twists of graded skew Clifford algebras of global dimension three or Ore extensions of graded Clifford algebras of global dimension two. Vancliff et al. \cite{VVe1, VVe2} go a step further and generalize the result of \cite{VVW}, between point modules and rank-two quadrics, by showing that point modules over GSCAs are determined by (noncommutative) quadrics of $\mu$-rank at most two.