The property of being a
quasi-triangular Hopf algebra is preserved by
twisting via an invertible element F = \sum_i f^i \otimes f_i \in \mathcal{A \otimes A} such that (\varepsilon \otimes id )F = (id \otimes \varepsilon)F = 1 and satisfying the cocycle condition : (F \otimes 1) \cdot (\Delta \otimes id)( F) = (1 \otimes F) \cdot (id \otimes \Delta)( F) Furthermore, u = \sum_i f^i S(f_i) is invertible and the twisted antipode is given by S'(a) = u S(a)u^{-1}, with the twisted comultiplication, R-matrix and co-unit change according to those defined for the
quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist. ==See also==