Double-graded quantum superplane

TL;DR

This paper introduces a new double-graded quantum superplane with a bicovariant calculus and demonstrates its Hopf algebra structure, expanding the framework of quantum supergeometry.

Contribution

It proposes the first $ ext{Z}_2 imes ext{Z}_2$-graded quantum superplane with explicit calculus and Hopf algebra properties, generalizing previous supergeometry models.

Findings

Constructed a bicovariant calculus on the double-graded quantum superplane.

Explicit commutation relations between coordinates, differentials, and derivatives.

Established a Hopf $ ext{Z}_2^2$-algebra structure for the extended superplane.

Abstract

A $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalisation of the quantum superplane is proposed and studied. We construct a bicovariant calculus on what we shall refer to as the \emph{double-graded quantum superplane}. The commutation rules between the coordinates, their differentials and partial derivatives are explicitly given. Furthermore, we show that an extended version of the double-graded quantum superplane admits a natural Hopf $\mathbb{Z}_2^2$-algebra structure.