Hom-coalgebra cleft extensions and Braided tensor Hom-categories of Hom-entwining Structures

Abstract

We investigate how the category of Hom-entwined modules can be made into a monoidal category. The sufficient and necessary conditions making the category of Hom-entwined modules have a braiding are given. Also, we formulate the concept of Hom-cleft extension for a Hom-entwining structure, and prove that if $(A, \alpha)$ is a $(C,\gamma)$-cleft extension, then there is an isomorphism of Hom-algebras between $(A, \alpha)$ and a crossed product Hom-algebra of $A^{coC}$ and $C$.