Hubble–Reynolds law

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

$R\mathrm{\Delta }(a)=\sigma \circ \mathrm{\Delta }(a)R,a\in \mathcal{𝒜}$

$\sigma :\mathcal{𝒜}\mathcal{\otimes }\mathcal{𝒜}\to \mathcal{𝒜}\mathcal{\otimes }\mathcal{𝒜}$

$x\otimes y\to y\otimes x$

so that $\sigma$ is the switch map and

$(\mathrm{\Delta }\otimes id)R={\mathrm{\Phi }}_{321}{R}_{13}{\mathrm{\Phi }}_{132}^{-1}{R}_{23}{\mathrm{\Phi }}_{123}$

$(id\otimes \mathrm{\Delta })R={\mathrm{\Phi }}_{231}^{-1}{R}_{13}{\mathrm{\Phi }}_{213}{R}_{12}{\mathrm{\Phi }}_{123}^{-1}$

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

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

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

• Quasitriangular Hopf algebra
• Ribbon Hopf algebra

References

• Vladimir Drinfeld, Quasi-Hopf algebras, Leningrad Math J. 1 (1989), 1419-1457
• J.M. Maillet and J. Sanchez de Santos, Drinfeld Twists and Algebraic Bethe Ansatz, Amer. Math. Soc. Transl. (2) Vol. 201, 2000