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 ) {\displaystyle {\mathcal {H_{A}}}=({\mathcal {A}},R,\Delta ,\varepsilon ,\Phi )} where B A = ( A , Δ , ε , Φ ) {\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )} is a quasi-Hopf algebra and R ∈ A ⊗ A {\displaystyle R\in {\mathcal {A\otimes A}}} known as the R-matrix, is an invertible element such that
R Δ ( a ) = σ ∘ Δ ( a ) R {\displaystyle R\Delta (a)=\sigma \circ \Delta (a)R}
for all a ∈ A {\displaystyle a\in {\mathcal {A}}} , where σ : A ⊗ A → A ⊗ A {\displaystyle \sigma \colon {\mathcal {A\otimes A}}\rightarrow {\mathcal {A\otimes A}}} is the switch map given by x ⊗ y → y ⊗ x {\displaystyle x\otimes y\rightarrow y\otimes x} , and
( Δ ⊗ id ) R = Φ 231 R 13 Φ 132 − 1 R 23 Φ 123 {\displaystyle (\Delta \otimes \operatorname {id} )R=\Phi _{231}R_{13}\Phi _{132}^{-1}R_{23}\Phi _{123}}
( id ⊗ Δ ) R = Φ 312 − 1 R 13 Φ 213 R 12 Φ 123 − 1 {\displaystyle (\operatorname {id} \otimes \Delta )R=\Phi _{312}^{-1}R_{13}\Phi _{213}R_{12}\Phi _{123}^{-1}}
where Φ a b c = x a ⊗ x b ⊗ x c {\displaystyle \Phi _{abc}=x_{a}\otimes x_{b}\otimes x_{c}} and Φ 123 = Φ = x 1 ⊗ x 2 ⊗ x 3 ∈ A ⊗ A ⊗ A {\displaystyle \Phi _{123}=\Phi =x_{1}\otimes x_{2}\otimes x_{3}\in {\mathcal {A\otimes A\otimes A}}} . The quasi-Hopf algebra becomes triangular if in addition, R 21 R 12 = 1 {\displaystyle R_{21}R_{12}=1} . The twisting of H A {\displaystyle {\mathcal {H_{A}}}} by F ∈ A ⊗ A {\displaystyle F\in {\mathcal {A\otimes 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 {\displaystyle \Phi =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.
See also Ribbon Hopf algebra
References Vladimir Drinfeld, "Quasi-Hopf algebras", Leningrad mathematical journal (1989), 1419–1457 J. M. Maillet and J. Sanchez de Santos, "Drinfeld Twists and Algebraic Bethe Ansatz", American Mathematical Society Translations: Series 2 Vol. 201, 2000
