A quasi-triangular quasi-Hopf algebra is a set \mathcal{H_A} = (\mathcal{A}, R, \Delta, \varepsilon, \Phi) where \mathcal{B_A} = (\mathcal{A}, \Delta, \varepsilon, \Phi) is a quasi-Hopf algebra and R \in \mathcal{A \otimes A} known as the R-matrix, is an invertible element such that
