New Turaev braided group categories and weak (co)quasi-Turaev group coalgebras

In order to construct a class of new braided crossed G-categories with nontrivial associativity and unit constraints, we study the G-graded monoidal category over a family of algebras {Hα}α∈G and introduce the notion of a weak (co)quasi-Turaev G-(co)algebra. Then we prove that the category of (co)representations of (co)quasitriangular weak (co)quasi-Turaev π-(co)algebras is exactly a braided crossed G-category. In fact, this (co)quasitriangular structure provides a solution to a generalized quantum Yang-Baxter type equation.