Classification, Koszulity and Artin–Schelter regularity of certain graded twisted tensor products

Abstract

Let $\mathbb{K}$ be an algebraically closed field. We classify all of the quadratic twisted tensor products $A{\otimes }_{\tau }B$ in the cases where $(A,B)=(\mathbb{K}[x],\mathbb{K}[y])$ and $(A,B)=(\mathbb{K}[x,y],\mathbb{K}[z])$. We determine when a quadratic twisted tensor product of this form is Koszul, and when it is Artin–Schelter regular.