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

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

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

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

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

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

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 $\mathrm{\Phi }=1$ 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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