Quantum Jordanian twist

The quantum deformation of the Jordanian twist F-qJ for the standard quantu m Borel algebra U-q (B) is constructed. It gives the family U-qJ (B) of qua ntum algebras depending on parameters xi and h. In a generic point these al gebras represent the hybrid (standard-nonstandard) quantization. The quantu m Jordanian twist can be applied to the standard quantization of any Kac-Mo ody algebra. The corresponding classical r-matrix is a linear combination o f the Drinfeld-Jimbo and the Jordanian ones. The two-parametric families of Hopf algebras obtained here are smooth and for the limit values of the par ameters the standard and nonstandard quantizations are recovered. The twist ing element F-qJ also has correlated limits; in particular when q tends to unity it acquires the canonical form. of the Jordanian twist. To illustrate the properties of the quantum Jordanian twist we construct the hybrid quan tizations for U(sl(2)) and fur the corresponding affine algebra U(<(sl(2))) over cap>. The universal quantum R-matrix and its defining representation a re presented.