Abstract
In this article, we first study the pre-T-category over a group π for a family of algebras {A α}α∈π and naturally introduce the notion of a quasi-Hopf π-coalgebra. Then we prove that the category of representations of quasitriangular quasi-Hopf π-coalgebras is exactly a Turaev braided π-category in the sense of Turaev. As a dual case, we also investigate the notion of a coquasitriangular coquasi-Hopf π-algebras and show that the category of corepresentations of coquasitriangular coquasi-Hopf π-algebras is also a Turaev braided π-category. Finally, we study the π-coring ℭ and the π-Doi-Hopf module for quasi-Hopf π-coalgebras, and show that the category of right-right π-Doi-Hopf modules is isomorphic to the category of right ℭ − π-comodules ℳπ−ℭ.
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