A quasi-triangular quasi-Hopf algebra is a specialized form of a quasi-Hopf algebra defined by the Ukrainian mathematician Vladimir Drinfeld in 1989. It is also a generalized form of a quasi-triangular Hopf algebra. A quasi-triangular quasi-Hopf algebra is a set where is a quasi-Hopf algebra and known as the R-matrix, is an invertible element such that for all , where is the switch map given by , and where and . The quasi-Hopf algebra becomes triangular if in addition, . The twisting of by is the same as for a quasi-Hopf algebra, with the additional definition of the twisted R-matrix