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.