Search

In order to construct a class of new Turaev-braided group category with nontrivial associativity, the concept of a quasitriangular quasi-Turaev group coalgebras was recently introduced. Inside the definition, the conditions of invertibility of the R-matrix R and bijectivity of the antipode S are required. In this article, we prove that the antipode of a quasitriangular quasi-Turaev group coalgebra without the assumptions about invertibility of the antipode and R-matrix is inner, and a fortiori, bijective. As an application, we prove that for a quasitriangular quasi-Turaev group coalgebra, two conditions mentioned above are unnecessary. [ABSTRACT FROM AUTHOR]

We consider two new algebras from anH-biquasimodule algebraAand a Hopf quasigroupH: twisted smash productA ⊛ Hand L-R smash productA⋇H, and find necessary and sufficient conditions for making them Hopf quasigroups. We generalize the main results in Brzeziński and Jiao [5] and Klim and Majid [9]. Moreover, ifHis a cocommutative Hopf quasigroup, we prove thatA ⊛ His isomorphic toA⋇Has Hopf quasigroups. [ABSTRACT FROM AUTHOR]

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]