n=3 Nilpotent differential calculus on some non-commutative (super) spaces

TL;DR

This paper develops covariant differential calculi of nilpotency order 2 and 3 on quantum planes and super quantum spaces, expanding the mathematical framework for non-commutative geometry with quantum symmetries.

Contribution

It introduces new covariant differential calculi with nilpotency orders 2 and 3 on quantum and super quantum spaces, generalizing previous differential calculus constructions.

Findings

Established differential calculi with d^2=0 and d^3=0 on quantum planes.

Constructed nilpotent differential calculi on super quantum spaces with quantum supergroup symmetries.

Extended the mathematical tools for non-commutative geometry and quantum group theory.

Abstract

In this paper, we construct a covariant differential calculus on quantum plane with two-parametric quantum group as a symmetry group. The two cases $d^2=0$ and $d^3=0$ are completly established. We also construct differential calculi $n=2$ and $n=3$ nilpotent on super quantum space with one and two-parametric symmetry quantum supergroup.