Algebraic Structures in Comodule Categories over Weak Bialgebras

Abstract: A classical result from the theory of Hopf algebras is that an H-module algebra is equivalent to an algebra object in the category of H-modules. We prove that we can significantly weaken some hypotheses on H to get a similar result.Here, we assume that H is a weak bialgebra: an algebra and coalgebra satisfying compatibility conditions weaker than those of a bialgebra. We show that there is an isomorphism of categories between H-comodule algebras and algebras in the category of comodules over H, as well as the analogous result for coalgebras and Frobenius algebras.