On the classification of graded twisted planes

Abstract

We use a representation of a graded twisted tensor product of $K[x]$ with $K[y]$ in $L(K^{{\mathbb{N}}_0})$ in order to obtain a nearly complete classification of these graded twisted tensor products via infinite matrices. There is one particular example and three main cases: quadratic algebras classified by Conner and Goetz, a family called $A(n,d,a)$ with the $n+1$-extension property for $n\ge 2$, and a third case, not fully classified, which contains a family $B(a,L)$ parameterized by quasi-balanced sequences.

In the fourth version we use a new method for the proof of Proposition 10.8, decomposing the infinite matrix $\tilde{M}$ into three (simpler) matrices. In the fifth version we shrten the article, and give a motivation for the representation in infinite matrices