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Let H be a quasi-Hopf algebra, a weak Hopf algebra or a braided Hopf algebra.Let B be an H-bicomodule algebra such that there exists a morphism ofH-bicomodule algebras v:H\rightarrow B. Then we can define an object B^{co(H)}which is a left-left Yetter-Drinfeld module over H, having extra propertiesthat allow to make a smash product B^{co(H)}# H which is an H-bicomodulealgebra, isomorphic to B.

Structure theorems for bicomodule algebras over quasi-Hopf algebras, weak Hopf algebras and braided Hopf algebras