Algebra of differential forms with exterior differential $d^3=0$ in dimension one

TL;DR

This paper constructs and analyzes a graded q-differential algebra of differential forms on a one-dimensional space where the exterior differential cubed equals zero, revealing a structure similar to anyonic line bimodules.

Contribution

It introduces a novel algebraic framework with a differential satisfying d^3=0, extending classical differential forms to a cubic root of unity setting.

Findings

The algebra is a graded q-differential algebra with q as a cubic root of unity.

Differential forms are generated by both first and second order differentials.

The bimodule structure resembles that of anyonic line bimodules.

Abstract

In this work, we construct the algebra of differential forms with the cube of exterior differential equal to zero on one-dimensional space. We prove that this algebra is a graded q-differential algebra where q is a cubic root of unity. Since the square of differential is not equal to zero the algebra of differential forms is generated not only by the first order differential but also by the second order differential of a coordinate. We study the bimodule generated by this second order differential, and show that its structure is similar to the structure of bimodule generated by the first order differential in the case of anyonic line.