Quasi-triangular quasi-Hopf algebra explained

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

l{H _{A}} = ( l{A}, R, \Delta, \varepsilon, \Phi) where l{B _{A}} = ( l{A}, \Delta, \varepsilon, \Phi) is a quasi-Hopf algebra and R \in l{A ⊗ A} known as the R-matrix, is an invertible element such that R \Delta(a) = \sigma \circ \Delta(a) R for all a \in l{A} , where \sigma\colon l{A ⊗ A} → l{A ⊗ A} is the switch map given by x ⊗ y → y ⊗ x , and(\Delta ⊗ \operatorname{id})R = \Phi _{231} R _{13} \Phi ^{-1} _{132} R _{23} \Phi _{123} (\operatorname{id} ⊗ \Delta)R = \Phi ^{-1} _{312} R _{13} \Phi _{213} R _{12} \Phi ^{-1} _{123}

where

\Phi _{abc} = x _{a} ⊗ x _{b} ⊗ x _{c} and \Phi _{123} = \Phi = x _{1} ⊗ x _{2} ⊗ x _{3} \in l{A ⊗ A ⊗ A} .

The quasi-Hopf algebra becomes triangular if in addition,

R _{21} R _{12} =1 .

The twisting of

l{H _{A}} by F \in l{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

\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