STRUCTURE THEOREMS FOR BICOMODULE ALGEBRAS OVER QUASI-HOPF ALGEBRAS, WEAK HOPF ALGEBRAS, AND BRAIDED HOPF ALGEBRAS

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 of H-bicomodule algebras v : H -> B. Then we can define an object B-co(H), which is a left-left Yetter-Drinfeld module over H, having extra properties that allow to make a smash product B-co(H)#H, which is an H-bicomodule algebra, isomorphic to B.