Twisting theory for weak Hopf algebras

Abstract

The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the (braided) monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl (2000). §1 IntroductionThe twisting theory of Hopf algebras introduced in [4] provides a way of obtaining new Hopf algebras from given ones using elements of the Hopf algebra cohomology theory. Drinfeld [4] obtained an equivalence between the (braided) monoidal categories of modules over the original and the twisted Hopf algebra, which was further investigated in [8] in the context of Hopf bimodules.Weak Hopf algebras (or quantum groupoids in the sense of [6]) are generalizations of Hopf algebras and groupoid algebras, and have been defined by [2]. The axioms are the same as the ones for a Hopf algebra, except that the coproduct of the unit, the product of the counit and the antipode condition are replaced by weaker properties.In this paper we mainly show that the results of twisting theory in [8] also hold true in the setting of weak Hopf algebra. Also the results in this paper hold true in the dual cases. This paper is organized as follows. In §2 we recall the notions of a weak Hopf algebra, weak Yetter-Drinfeld module, and a weak Hopf bimodule needed in the sequel. In §3 we study the twisting theory of weak Hopf algebras that generalizes the result from [8]. In §4 we give an equivalence between the (braided) monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra.

Sovereign and ribbon weak Hopf algebras

“…A coquasitriangular weak Hopf algebra ( [5]) ðH; sÞ consists of a weak Hopf algebra H and a map s : H n H ! k satisfying the following conditions:…”