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

:$R\; Delta(a)\; =\; sigma\; circ\; Delta(a)\; R,\; a\; in\; mathcal\{A\}$:$sigma:\; mathcal\{A\; otimes\; A\}\; ightarrow\; mathcal\{A\; otimes\; A\}$:$x\; otimes\; y\; ightarrow\; y\; otimes\; x$

so that $sigma$ is the switch map and

:$(Delta\; otimes\; id)R\; =\; Phi\_\{321\}R\_\{13\}Phi\_\{132\}^\{-1\}R\_\{23\}Phi\_\{123\}$:$(id\; otimes\; Delta)R\; =\; Phi\_\{231\}^\{-1\}R\_\{13\}Phi\_\{213\}R\_\{12\}Phi\_\{123\}^\{-1\}$

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

*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