Weak Group Entwined Modules And Generalized Cibils-Rosso's Theorem

We introduce the notions of a weak group entwining structure and a weak group smash product as generalizations of Caenepeel and Groot (see[15]). We prove that there is one-one correspondence between weak group entwining structures and weak group smash product structures and show that a generalized Cibils-Rosso's theorem holds in the setting of weak group entwining structures, that is, the weak group entwined bimodules are modules over a certain group graded algebra. As applications, we consider the our theories in the setting of (weak) Hopf group coalgebras.