Relative Hom-Hopf modules and total integrals

TL;DR

This paper explores the existence of total integrals in monoidal Hom-Hopf algebras, establishing criteria for their existence and implications for category equivalences, generalizing classical results.

Contribution

It introduces new criteria for total integrals in monoidal Hom-Hopf algebras and links them to category equivalences, extending Doi's classical results.

Findings

Existence of total integrals characterized by injectivity of representations.

Introduction of total quantum integrals and their role in category equivalences.

Establishment of an affineness criterion involving surjective canonical maps.

Abstract

Let $(H, \a)$ be a monoidal Hom-Hopf algebra and $(A, \b)$ a right $(H, \a)$-Hom-comodule algebra. We first investigate the criterion for the existence of a total integral of $(A, \b)$ in the setting of monoidal Hom-Hopf algebras. Also we prove that there exists a total integral $\phi: (H, \a)\rightarrow (A, \b)$ if and only if any representation of the pair $(H,A)$ is injective in a functorial way, as a corepresentation of $(H, \a)$, which generalizes Doi's result. Finally, we define a total quantum integral $\g: H\rightarrow Hom(H, A)$ and prove the following affineness criterion: if there exists a total quantum integral $\g$ and the canonical map $\psi: A\o_{B}A\rightarrow A\o H,\ \ a\o_{B}b\mapsto \b^{-1}(a)b_{[0]}\o \a(b_{[1]}) $is surjective, then the induction functor $A\o_B-: \widetilde{\mathscr{H}}(\mathscr{M}_k)_{B}\rightarrow \widetilde{\mathscr{H}}(\mathscr{M}_k)^{H}_{A}$ is an equivalence of categories.