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

, ε ,

, ε ,

is a quasi-Hopf algebra and

known as the R-matrix, is an invertible element such that for all

is the switch map given by

The quasi-Hopf algebra becomes triangular if in addition,

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.

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