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
so that is the switch map 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 (resp. triangular) quasi-Hopf algebra with 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.
Famous quotes containing the word algebra:
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)