Quantum projective planes as certain graded twisted tensor products

TL;DR

This paper classifies quadratic graded twisted tensor products of polynomial algebras, identifies their geometric properties when regular, and explores their relation to Sklyanin algebras and quantum projective planes.

Contribution

It provides a classification of quadratic graded twisted tensor products and characterizes their geometric and algebraic properties, including connections to Sklyanin algebras.

Findings

Classification of quadratic graded twisted tensor products

Identification of point schemes for Artin-Schelter regular cases

Description of subalgebras within three-dimensional Sklyanin algebras

Abstract

Let $\mathbb{k}$ be an algebraically closed field. Building upon previous work, we classify, up to isomorphism of graded algebras, quadratic graded twisted tensor products of $\mathbb{k}[x,y]$ and $\mathbb{k}[z]$. When such an algebra is Artin-Schelter regular, we identify its point scheme and type. We also describe which three-dimensional Sklyanin algebras contain a subalgebra isomorphic to a quantum $\mathbb{P}^1$, and we show that every algebra in this family is a graded twisted tensor product of $\mathbb{k}_{-1}[x,y]$ and $\mathbb{k}[z]$.