Right-covariant differential calculus on ${\cal F}({\mathbb C}_q^{2|1})$

TL;DR

This paper introduces a new ${f Z}_2$-graded quantum space, constructs a right-covariant differential calculus on it, and explores its algebraic structures including a quantum Weyl algebra and its dual Hopf algebra.

Contribution

It defines a novel ${f Z}_2$-graded quantum space and develops a compatible differential calculus, expanding the algebraic framework of quantum superspaces.

Findings

The algebra of polynomials forms a ${f Z}_2$-graded Hopf algebra.

A right-covariant differential calculus is constructed on the quantum space.

The dual ${f Z}_2$-graded Hopf algebra is explicitly realized.

Abstract

We define a new ${\mathbb Z}_2$-graded quantum (2+1)-space and show that the extended ${\mathbb Z}_2$-graded algebra of polynomials on this ${\mathbb Z}_2$-graded quantum space, denoted by ${\cal F}({\mathbb C}_q^{2\vert1})$, is a ${\mathbb Z}_2$-graded Hopf algebra. We construct a right-covariant differential calculus on ${\cal F}({\mathbb C}_q^{2\vert1})$ and define a ${\mathbb Z}_2$-graded quantum Weyl algebra and mention a few algebraic properties of this algebra. Finally, we explicitly construct the dual ${\mathbb Z}_2$-graded Hopf algebra of ${\cal F}({\mathbb C}_q^{2\vert1})$.