Integrals for monoidal Hom-Hopf algebras and their applications

Source abstract

Integrals of monoidal Hom-Hopf algebras are introduced and the existence and uniqueness of integrals for finite-dimensional monoidal Hom-Hopf algebras are investigated first. Then integrals are applied to the Maschke type theorem for monoidal Hom-Hopf algebras controlling the semisimplicity and separability of monoidal Hom-Hopf algebras. Further, monoidal Hom-algebras are characterized with additional Frobenius property, and the question when finite-dimensional monoidal Hom-Hopf algebras are Frobenius is studied. As applications of integrals, the Maschke type theorem for Hom-smash product is given, and the Morita context in the Hom-category \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathcal {H}}(\mathcal {M}_k)$\end{document}H̃(Mk) is constructed.