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 $\mathcal = (\mathcal, R, \Delta, \varepsilon, \Phi)$ where $\mathcal = (\mathcal, \Delta, \varepsilon, \Phi)$ is a quasi-Hopf algebra and $R \in \mathcal$ known as the R-matrix, is an invertible element such that
:$R \Delta(a) = \sigma \circ \Delta(a) R$
for all $a \in \mathcal$, where $\sigma\colon \mathcal \rightarrow \mathcal$ is the switch map given by $x \otimes y \rightarrow y \otimes x$, and

:$(\Delta \otimes \operatorname)R = \Phi_R_\Phi_^R_\Phi_$
:$(\operatorname \otimes \Delta)R = \Phi_^R_\Phi_R_\Phi_^$

where $\Phi_ = x_a \otimes x_b \otimes x_c$ and $\Phi_= \Phi = x_1 \otimes x_2 \otimes x_3 \in \mathcal$.

The quasi-Hopf algebra becomes ''triangular'' if in addition, $R_R_=1$.

The twisting of $\mathcal$ by $F \in \mathcal$ 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

Coalgebras

{{abstract-algebra-stub