# Covariant Differential Calculus Over Monoidal Hom-Hopf Algebras

## TL;DR

This paper develops a covariant differential calculus framework over monoidal Hom-Hopf algebras, extending classical concepts to the Hom-algebra setting and establishing quantum analogues of Lie algebra structures.

## Contribution

It introduces the first order differential calculus on monoidal Hom-algebras, extends universal calculus, and defines quantum Hom-tangent spaces with Lie bracket analogues.

## Key findings

- Defined covariant FODC over monoidal Hom-Hopf algebras.
- Extended universal FODC to Hom-differential calculus.
- Proved quantum antisymmetry and Hom-Jacobi identities.

## Abstract

Concepts of first order differential calculus (FODC) on a monoidal Hom-algebra and left-covariant FODC over a left Hom-quantum space with respect to a monoidal Hom-Hopf algebra are presented. Then, extension of the universal FODC over a monoidal Hom-algebra to a universal Hom-differential calculus is described. Next, concepts of left(right)-covariant and bicovariant FODC over a monoidal Hom-Hopf algebra are studied in detail. Subsequently, notion of quantum Hom-tangent space associated to a bicovariant Hom-FODC is introduced and equipped with an analogue of Lie bracket (commutator) through Woronowicz' braiding. Finally, it is proven that this commutator satisfies quantum versions of the antisymmetry relation and Hom-Jacobi identity.