A structure theorem for quasi-Hopf comodule algebras

Abstract

If H is a quasi-Hopf algebra and B is a right H-comodule algebra such that there exists v: H → B a morphism of right H-comodule algebras, we prove that there exists a left H-module algebra A such that B ≃ A#H. The main difference when comparing to the Hopf case is that, from the multiplication of B, which is associative, we have to obtain the multiplication of A, which in general is not; for this we use a canonical projection E arising from the fact that B becomes a quasi-Hopf H-bimodule.