A differential calculus on Z$_3$-graded quantum superspace ${\mathbb R}_q(2|1)$

TL;DR

This paper develops a new Z3-graded quantum superspace, constructs a differential calculus on it, and identifies a corresponding Z3-graded quantum supergroup, expanding the algebraic framework of quantum superspaces.

Contribution

It introduces a Z3-graded quantum superspace, defines its Hopf algebra structure, and constructs a compatible differential calculus and supergroup, which are novel in the field.

Findings

Defined a Z3-graded quantum superspace and its Hopf algebra structure.

Constructed a differential calculus on the superspace.

Discovered a new Z3-graded quantum supergroup.

Abstract

We introduce a Z$_3$-graded quantum $(2+1)$-superspace and define Z$_3$-graded Hopf algebra structure on algebra of functions on the Z$_3$-graded quantum superspace. We construct a differential calculus on the Z$_3$-graded quantum superspace, and obtain the corresponding Z$_3$-graded Lie superalgebra. We also find a new Z$_3$-graded quantum supergroup which is a symmetry group of this calculus.