In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of such that * for all , where is the coproduct on H, and the linear map is given by , * , * , where , , and , where , , and , are algebra morphisms determined by R is called the R-matrix. As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang-Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity, ; moreover , , and where (cf. Ribbon Hopf algebras).