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 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 quasi-Hopf algebra with is a quasi-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.
