Quasi-triangular quasi-Hopf algebra

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 formula_1 where formula_2 is a quasi-Hopf algebra and formula_3 known as the R-matrix, is an invertible element such that 
for all formula_5, where formula_6 is the switch map given by formula_7, and
where formula_10 and formula_11.
The quasi-Hopf algebra becomes "triangular" if in addition, formula_12.
The twisting of formula_13 by formula_14 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 formula_15 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.