Point modules over regular graded skew Clifford algebras

TL;DR

This paper extends the classification of point modules from regular graded Clifford algebras to graded skew Clifford algebras, showing they are determined by noncommutative quadrics of rank at most two within the associated noncommutative quadric system.

Contribution

It generalizes previous results on point modules from GCA to GSCA using the concept of {00}-rank, broadening the understanding of noncommutative algebraic structures.

Findings

Point modules over regular GSCAs are characterized by noncommutative quadrics of {00}-rank at most two.

The notion of {00}-rank is key to extending classical results to noncommutative settings.

The results unify the understanding of point modules across commutative and noncommutative graded algebras.

Abstract

Results of Vancliff, Van Rompay and Willaert in 1998 prove that point modules over a regular graded Clifford algebra (GCA) are determined by (commutative) quadrics of rank at most two that belong to the quadric system associated to the GCA. In 2010, Cassidy and Vancliff generalized the notion of a GCA to that of a graded skew Clifford algebra (GSCA). The results in this article show that prior results may be extended, with suitable modification, to GSCAs. In particular, using the notion of {\mu}-rank introduced recently by the authors, the point modules over a regular GSCA are determined by (noncommutative) quadrics of {\mu}-rank at most two that belong to the noncommutative quadric system associated to the GSCA.