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Quantum integrals associated to quantum Hom-Yetter-Drinfel’d modules are defined, and the affineness criterion for quantum Hom-Yetter-Drinfel’d modules is proved in the following form. Let (H,α) be a monoidal Hom-Hopf algebra, (A,β) an (H,α)-Hom-bicomodule algebra and . Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map , , is surjective, we prove that the induction functor is an equivalence of categories.
We continue our study of the category of Doi Hom-Hopf modules introduced in [Colloq. Math., to appear]. We find a sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. We also obtain a condition for a monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Moreover, we introduce morphisms between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras, which give rise to functors between the category of Doi Hom-Hopf...
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