Quantum Jordanian twist
Vladimir Lyakhovsky, Alexandr Mirolubov, Mariano del Olmo
Abstract
The quantum deformation of the Jordanian twist FqJ for the standard quantum Borel algebra Uq(B) is constructed. It gives the family UqJ(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 FqJ 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.
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