2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/150828We 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 $D^ϕ(G)$ associated to a finite group $G$ and group 3-cocycle $ϕ$.LATEX 9 pages, final ver. (bit more explicit) as accepted Lett.Math.PhysQuantum Algebratext