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 ${\mathcal {H_{A}}}=({\mathcal {A}},R,\Delta ,\varepsilon ,\Phi )$ where ${\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )$ is a quasi-Hopf algebra and $R\in {\mathcal {A\otimes A}}$ known as the R-matrix, is an invertible element such that

R\Delta (a)=\sigma \circ \Delta (a)R

for all $a\in {\mathcal {A}}$ , where $\sigma \colon {\mathcal {A\otimes A}}\rightarrow {\mathcal {A\otimes A}}$ is the switch map given by $x\otimes y\rightarrow y\otimes x$ , and

(\Delta \otimes \operatorname {id} )R=\Phi _{231}R_{13}\Phi _{132}^{-1}R_{23}\Phi _{123}

(\operatorname {id} \otimes \Delta )R=\Phi _{312}^{-1}R_{13}\Phi _{213}R_{12}\Phi _{123}^{-1}

where $\Phi _{abc}=x_{a}\otimes x_{b}\otimes x_{c}$ and $\Phi _{123}=\Phi =x_{1}\otimes x_{2}\otimes x_{3}\in {\mathcal {A\otimes A\otimes A}}$ .

The quasi-Hopf algebra becomes triangular if in addition, $R_{21}R_{12}=1$ .

The twisting of ${\mathcal {H_{A}}}$ by $F\in {\mathcal {A\otimes A}}$ 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 $\Phi =1$ 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.

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