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

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