Some ribbon elements for the quasi-Hopf algebra $D^{\omega }(H)$

We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra $D^{\omega }(H)$ defined in \cite{bp} and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for $D^{\omega }(H)$.