The fundamental theorem and Maschke’s theorem in the category of relative Hom-Hopf modules

We introduce the concept of relative Hom-Hopf modules and investigate their structure in a monoidal category (H) over tilde (M-k). More particularly, the fundamental theorem for relative Hom-Hopf modules is proved under the assumption that the Hom-comodule algebra is cleft. Moreover, Maschke's theorem for relative Hom-Hopf modules is established when there is a multiplicative total Hom-integral.