The Gervais-Neveu-Felder equation for the Jordanian quasi-Hopf Uh;y(sl(2)) algebra

Using a contraction procedure, we construct a twist operator that satisfies a shifted cocycle condition, and leads to the Jordanian quasi-Hopf Uh;y(sl(2)) algebra. The corresponding universal Rh(y) matrix obeys a Gervais-Neveu-Felder equation associated with the Uh;y(sl(2)) algebra. For a class of representations, the dynamical Yang-Baxter equation may be expressed as a compatibility condition for the algebra of the Lax operators. [email protected] Laboratoire Propre du CNRS UPR A.0014 Recently a class of invertible maps between the classical sl(2) and the non-standard Jordanian Uh(sl(2)) algebras has been obtained [1]-[3]. The classical and the Jordanian coalgebraic structures may be related [2]-[5] by the twist operators corresponding to these maps. Following the first twist leading from the classical to the Jordanian Hopf structure, it is possible to envisage a second twist leading to a quasi-Hopf quantization of the Jordanian Uh(sl(2)) algebra. By explicitly constructing the appropriate universal twist operator that satisfies a shifted cocycle condition, we here obtain the Gervais-Neveu-Felder (GNF) equation satisfied by the universal R matrix of a one-parametric quasi-Hopf deformation of the Uh(sl(2)) algebra. The GNF equation corresponding to the standard Drinfeld-Jimbo deformed Uq(sl(2)) algebra was studied in the context of Liouville field theory [6], quantization of KniznikZamolodchikov-Bernard equation [7] and the quantization of the Calogero-Moser model in the R matrix formalism [8]. The general construction of the twist operators leading to the GNF equation corresponding to the quasi-triangular standard Drinfeld-Jimbo deformed Uq(g) algebras and superalgebras were obtained in [9]-[11]. For the sake of completeness, we start by enlisting the general properties of a quasi-Hopf algebra A [12]. For all a ∈ A there exist an invertible element Φ ∈ A ⊗ A ⊗ A and the elements (α, β) ∈ A, such that (id⊗△)△ (a) = Φ (△⊗ id)(△(a)) Φ, (id⊗ id⊗△)(Φ) (△⊗ id⊗ id)(Φ) = (1⊗ Φ) (id⊗△⊗ id)(Φ) (Φ⊗ 1), (ε⊗ id) ◦ △ = id, (id⊗ ε) ◦ △ = id,