Invertible skew pairings and crossed products for weak Hopf algebras

Abstract

In this paper we work with invertible skew pairings for weak bialgebras in a symmetric monoidal category where every idempotent morphism splits. We prove that this kind of skew pairings induces examples of weak distributive laws and therefore they provide weak wreath products. Also we will show that they define weakly comonoidal mutually weak inverse pairs of weak distributive laws and, by the results proved by G. Böhm and J. Gómez-Torrecillas, we obtain weak wreath products that become weak bialgebras with respect to the tensor product coalgebra structure. As an application, we will show that the Drinfel'd double of a finite weak Hopf algebra can be constructed using the weak wreath product associated to an invertible $1$-skew pairing.