Yetter–Drinfeld category for the quasi-Turaev group coalgebra and cocycle deformation

Abstract

Let [Formula: see text] be a group and [Formula: see text] a quasi-Turaev group coalgebra over [Formula: see text]. In this paper, we firstly construct the category [Formula: see text] of left-right Yetter–Drinfeld modules over [Formula: see text], generalizing both of the Yetter–Drinfeld modules over Hopf group coalgebra and quasi-Hopf algebra, and prove that this category is isomorphic to the center of the representation category of [Formula: see text]. Next, we prove that the full subcategory [Formula: see text] consisting of all finite dimensional Yetter–Drinfeld modules over [Formula: see text] is autonomous. Finally, when [Formula: see text] is reduced to a Turaev group coalgebra, we introduce the dual 2-cocycle [Formula: see text] of [Formula: see text], and dual cocycle deformation [Formula: see text], and show that [Formula: see text] is isomorphic to [Formula: see text] as braided monoidal category.