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 A quasi-triangular (resp. triangular) quasi-Hopf algebra with is a quasi-triangular (resp. triangular) Hopf algebra as the latter two conditions in the definition reduce the conditions of quasi-triangularity of a Hopf algebra. Similarly to the twisting properties of the quasi-Hopf algebra, the property of being quasi-triangular or triangular quasi-Hopf algebra is preserved by twisting. See also Ribbon Hopf algebra References Vladimir Drinfeld, "Quasi-Hopf algebras", Leningrad mathematical journal (1989), 1419–1457 J. M. Maillet and J. Sanchez de Santos, "Drinfeld Twists and Algebraic Bethe Ansatz", American Mathematical Society Translations: Series 2 Vol. 201, 2000 Coalgebras