Quantum de Rham complex with $d^3 = 0$ differential

TL;DR

This paper develops a quantum de Rham complex with a third-order nilpotent differential operator, exploring its algebraic structure and implications for quantum planes with varied bosonic or fermionic properties.

Contribution

It introduces a novel quantum de Rham complex with a third-order differential operator satisfying $d^3=0$, extending classical differential calculus to quantum algebra settings.

Findings

Second order differentials generate bosonic or fermionic quantum planes

Construction of de Rham complex with $d^3=0$ on quantum algebra

Analysis of differential structures depending on parameter Q

Abstract

In this work, we construct the de Rham complex with differential operator d satisfying the Q-Leibniz rule, where Q is a complex number, and the condition $d^3=0$ on an associative unital algebra with quadratic relations. Therefore we introduce the second order differentials $d^2x^i$. In our formalism, besides the usual two-dimensional quantum plane, we observe that the second order differentials $d^2 x$ and $d^2 y$ generate either bosonic or fermionic quantum planes, depending on the choice of the differentiation parameter Q.