New Turaev Braided Group Categories and Group Corings Based on Quasi-Hopf Group Coalgebras

Yetter-Drinfeld category for the quasi-Turaev group coalgebra

“…Let π be a group with the unit 1. Recall from [1] that a crossed category C (over π) is given by the following data:…”

“…Recall from [1], a family of algebras H = {H α } α∈π is a quasi-semi-T-coalgebra if there exist a family of morphisms of algebra ∆ = {∆ α,β :…”

“…M. Zunino in [11] constructed the Yetter-Drinfeld category of crossed Hopf group coalgebra and showed that it is a Turaev braided group category. Motivated by this construction, in this paper, we will generalize this result to quasi-Turaev Hopf group coalgebra defined in [1]. The notion of Yetter-Drinfeld category of quasi-Turaev Hopf group coalgebra will be given, and the isomorphism between Yetter-Drinfeld category and the category of the center of representation category of quasi-Turaev Hopf group coalgebra will be established.…”

“…Based on the work of [4], more results have been obtained in [2] and [9], where the method used in [4] was applied to weak Hopf algebras and regular multiplier Hopf algebras. It is well-known that there is anther approach to the construction, for instance, in [1] the authors introduced the notion of quasi-Hopf group coalgebras and proved that the representation category of quasitriangular quasi-Hopf group coalgebras is exactly a Turaev braided group category. M. Zunino in [11] constructed the Yetter-Drinfeld category of crossed Hopf group coalgebra and showed that it is a Turaev braided group category.…”

The construction of braided $T$-category via Yetter-Drinfeld-Long bimodules

“…One is to construct Turaev group-coalgebra, the representative category of which is a braided T -category. For details the readers can refer to [1,15,18]. The other was proposed by F. Panaite and M. Staic in their paper [8], which is based on some kind of generalized Yetter-Drinfeld module of Hopf algebra with bijective antipode.…”