The Drinfel'd Double And The C.M.Z.-Theorem For WT-Coalgebras

Let π be a group. In this paper, we introduce the notions of a weak Doi-Hopf π-module and a weak π-twisted smash product. We show that the Yetter-Drinfel'd π-modules over a weak crossed Hopf π-coalgebra (WT-coalgebra) are special cases as these new weak Doi-Hopf π-modules, generalizing the main result by Caenepeel et al. (reference[3]) and that the Drinfel'd double for WT-coalgebras (reference[17]) appears as a type of such a weak π-twisted smash product, respectively. Finally, starting from a weak Hopf algebra endowed with an action of a group π by weak Hopf automorphisms, we construct a quasitriangular weak Hopf π-coalgebra by a twisted double method, generalizing the main result in Virelizier (reference[19]). This method allows us to obtain non-trivial examples of quasitriangular weak Hopf π-coalgebras.