Quantum Double for QuasiHopf Algebras

We introduce a quantum double quasitriangular quasi-Hopf algebra $D(H)$ associated to any quasi-Hopf algebra $H$. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double $D^\phi(G)$ associated to a finite group $G$ and group 3-cocycle $\phi$.