Isomorphisms of graded skew Clifford algebras

Abstract

A regular algebra of global dimension $n+1$ is often called a quantum ${\mathbb{P}}^n$. In 2011, Nafari, Vancliff and Zhang showed that graded skew Clifford algebras (GSCAs) could be used to classify most quadratic quantum ${\mathbb{P}}^2$s. Some time later, Chandler, Tomlin and Vancliff used their work with certain families of GSCAs to develop a conjecture on the quantum space of a generic quadratic quantum ${\mathbb{P}}^3$. These results suggest that (the isomorphism classes of) GSCAs are likely to play a fundamental role in the classification of quadratic quantum ${\mathbb{P}}^3$s. In this article, we will discuss some of these results on GSCAs and discuss new results on isomorphisms between GSCAs.