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

${\displaystyle R\Delta (a)=\sigma \circ \Delta (a)R}$

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

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

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

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

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

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