# On the classification of graded twisted planes

## TL;DR

This paper classifies graded twisted tensor products of polynomial algebras using matrix representations, covering known quadratic cases, a new family with extension properties, and a partially classified family involving quasi-balanced sequences.

## Contribution

It provides a nearly complete classification of graded twisted tensor products of polynomial algebras through matrix methods, including new families and extending previous classifications.

## Key findings

- Classified quadratic algebras by Conner and Goetz.
- Identified a new family $A(n,d,a)$ with extension properties.
- Described a partially classified family $B(a,L)$ with quasi-balanced sequences.

## Abstract

We use a representation of a graded twisted tensor product of $K[x]$ with $K[y]$ in $L(K^{\Bbb{N}_0})$ 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.