CROSSED PRODUCTS AND CLEFT EXTENSIONS FOR COQUASI-HOPF ALGEBRAS

The notion of crossed product by a coquasi-bialgebra H is introduced and studied. This is an algebra in the monoidal category of right H-comodules. We give an interpretation of the crossed product as an action of a monoidal category. In particular, necessary and sufficient conditions for two crossed products to be equivalent are provided. Then, two structure theorems for coquasi-Hopf modules are given. First, these are relative Hopf modules over the crossed product. Second, the category of coquasi-Hopf modules is trivial, namely equivalent to the category of modules over the starting associative algebra. In connection with the crossed product, we recall from [1] the notion of a cleft extension over a coquasi-Hopf algebra. A Morita context of Hom spaces is constructed in order to explain these extensions, which are shown to be equivalent to crossed products with invertible cocycles. At the end, we give a complete description of all cleft extensions over the non-trivial coquasi- Hopf algebras of dimension two and three.