  The quantum deformation of the Jordanian twist F_qJ for the standard quantum
Borel algebra U_q(B) is constructed. It gives the family U_qJ(B) of quantum
algebras depending on parameters x and h. In a generic point these algebras
represent the hybrid (standard-nonstandard) quantization. The quantum Jordanian
twist can be applied to the standard quantization of any Kac-Moody algebra. The
corresponding classical r-matrix is a linear combination of the Drinfeld- Jimbo
and the Jordanian ones. The obtained two-parametric families of Hopf algebras
are smooth and for the limit values of the parameters the standard and
nonstandard quantizations are recovered. The twisting element F_qJ also has the
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 quantizations for U(sl(2)) and
for the corresponding affine algebra U(hat(sl(2))). The universal quantum
R-matrix and its defining representation are presented.
