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, a \in \mathcal{A}
\sigma: \mathcal{A \otimes A} \rightarrow \mathcal{A \otimes A}
x \otimes y \rightarrow y \otimes x

so that \sigma is the switch map and

(\Delta \otimes \operatorname{id})R = \Phi_{321}R_{13}\Phi_{132}^{-1}R_{23}\Phi_{123}
(\operatorname{id} \otimes \Delta)R = \Phi_{231}^{-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.

See also

References


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