Skew Clifford Algebras

TL;DR

This paper introduces skew Clifford algebras as a generalization of classical Clifford algebras, explores their homogenizations, and connects them to Artin-Schelter regular algebras, expanding the understanding of graded algebra deformations.

Contribution

It defines skew Clifford algebras, relates them to graded skew Clifford algebras, and investigates their homogenizations and conditions for Artin-Schelter regularity.

Findings

Skew Clifford algebras are the $ ext{Z}_2$-graded PBW deformations of quantum exterior algebras.

Certain skew Clifford algebras can be homogenized to form Artin-Schelter regular algebras.

The paper determines the possible dimensions of skew Clifford algebras.

Abstract

We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincar\' e-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the $\mathbb{Z}_2$-graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples.