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 $B = A^{coH}$. Under the assumption that there exists a total quantum integral γ: H → Hom(H,A) and the canonical map $β: A ⊗_{B} A → A ⊗ H$, $a ⊗_{B} b↦ S^{-1}(b_{[1]})α(b_{[0][-1]}) ⊗ β^{-1}(a)β(b_{[0][0]})$, is surjective, we prove that the induction functor $A ⊗_{B}-: 𝓗̃ (𝓜 _{k})_{B} → ^{H}𝓗 𝓨 𝓓_{A}$ is an equivalence of categories.
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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 modules, and we study tensor identities for monodial categories of Doi Hom-Hopf modules. Furthermore, we construct a braiding on the category of Doi Hom-Hopf modules. Finally, as an application of our theory, we get a braiding on the category of Hom-modules, on the category of Hom-comodules, and on the category of Hom-Yetter-Drinfeld modules.
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