Given a bialgebra , a Drinfel’d twist is an invertible element satisfying
(1\otimes\chi)(id\otimes\Delta)\chi = (\chi\otimes 1)(\Delta\otimes id)\chi,
Vladimir Drinfel’d introduced in order to suggest a procedure of twisting Hopf algebras. Given a bialgebra, Hopf algebra or quasitriangular Hopf algebra, one gets another such structure twisting it with a Drinfel’d twist. Let be a quasi-triangular Hopf algebra with with comultiplication , antipode , universal R-element and let be a Drinfel’d twist. Then the same algebra becomes another quasitriangular Hopf algebra with formulas for comultiplication , universal R-element and antipode . The counit is not changed. Here where is the flip of tensor factors, and is a notation similar to Sweedler’s convention.