Relative Hom-Hopf modules and total integrals

Abstract

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) a right (H, α)-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of (A, β) in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral ϕ : (H, α) → (A, β) if and only if any representation of the pair (H, A) is injective in a functorial way, as a corepresentation of (H, α), which generalizes Doi’s result. Finally, we define a total quantum integral γ : H → Hom(H, A) and prove the following affineness criterion: if there exists a total quantum integral γ and the canonical map ψ : A⊗BA → A ⊗ H, a⊗Bb ↦ β−1(a) b[0] ⊗ α(b[1]) is surjective, then the induction functor A⊗B−:ℋ˜(ℳk)B→ℋ˜(ℳk)AH is an equivalence of categories.