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

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.