Integrals for monoidal Hom-Hopf algebras and their applications

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 (H) over tilde (M-k) is constructed. (C) 2013 AIP Publishing LLC.