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

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

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

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

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

where ${\Phi }_{abc}=x_a\otimes x_b\otimes x_c$ and ${\Phi }_{123}=\Phi =x_1\otimes x_2\otimes x_3\in 𝒜\mathcal{\otimes }𝒜\mathcal{\otimes }𝒜$.

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

The twisting of ${\mathscr{H}}_{𝒜}$ by $F\in 𝒜\mathcal{\otimes }𝒜$ 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

• 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

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