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The notion of a group C-ring is introduced. Furthermore, the relationships of weak group entwining structures, invertible weak group entwining structures and group C-rings are investigated, which generalize the recent work of Brzezinski [5] and Brzezinski and Turner [11]. As applications, we finally construct two kinds of examples of weak entwining structures by weak group algebra Galois coextensions and weak Hopf group algebras, respectively. [ABSTRACT FROM AUTHOR]

We show that the antipode of a braided dual quasi-Hopf algebra is inner, and, a fortiori, bijective. This improves a result of Li [10]. [ABSTRACT FROM AUTHOR]

The notions of a “weak group entwining structure” and an “invertible weak group entwining structure” are introduced. Furthermore, the relationship between them and group corings is also investigated. It turns out that (weak) group entwining structures and (weak) group entwined modules can be viewed as group corings and comodules over group corings, respectively, which generalizes the recent work of Brzezinski (2002) and Brzezinski et al. (2006). [ABSTRACT FROM AUTHOR]