Quasi-triangular Quasi-Hopf algebra

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


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