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 H A = ( A , R , Δ , ε , Φ ) where B A = ( A , Δ , ε , Φ ) is a quasi-Hopf algebra and R ∈ A ⊗ A known as the R-matrix, is an invertible element such that

so that σ is the switch map and

where Φ a b c = x a ⊗ x b ⊗ x c and Φ 123 = Φ = x 1 ⊗ x 2 ⊗ x 3 ∈ A ⊗ A ⊗ A .

The quasi-Hopf algebra becomes triangular if in addition, R 21 R 12 = 1 .

The twisting of H A by F ∈ A ⊗ A 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 Φ = 1 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.