Hom-coalgebra cleft extensions and braided tensor Hom-categories of Hom-entwining structures

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 $C$.

Keywords:

Hom-Hopf algebra Hom-entwining structure, cleft extension,

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