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

Abstract

This paper is a survey of work done on $\mathbb {Z}$-graded Clifford algebras (GCAs) and $\mathbb {Z}$-graded skew Clifford algebras (GSCAs) by Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998), Stephenson and Vancliff (J Algebra 312(1): 86–110, 2007), Cassidy and Vancliff (J Lond Math Soc 81:91–112, 2010), Nafari et al. (J Algebra 346(1): 152–164, 2011), Vancliff and Veerapen (Contemp Math 592: 241–250, 2013), (J Algebra 420:54–64, 2014). In particular, we discuss the hypotheses necessary for these algebras to be Artin Schelter-regular (Adv Math 66:171–216, 1987), (The Grothendieck Festschrift. Birkhäuser, Boston, 1990) 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 Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998) 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 Stephenson and Vancliff (J Algebra 312(1): 86–110, 2007) provide a method to construct GCAs with finitely many distinct isomorphism classes of point modules. In Cassidy and Vancliff (J Lond Math Soc 81:91–112, 2010), Cassidy and Vancliff introduce a quantized analog of a GCA, called a graded skew Clifford algebra and Nafari et al. (J Algebra 346(1): 152–164, 2011) 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 and Veerapen (Contemp Math 592: 241–250, 2013), (J Algebra 420:54–64, 2014) go a step further and generalize the result of Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998), 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.