Interpolations between Jordanian twists, the Poincaré-Weyl algebra and dispersion relations

Abstract: We consider a two parameter family of Drinfeld twists generated from a simple Jordanian twist further twisted by 1-cochains. Twists from this family interpolate between two simple Jordanian twists. Relations between them are constructed and discussed. It is proved that there exists a one parameter family of twists identical to a simple Jordanian twist. The twisted coalgebra, star product and coordinate realizations of the $\kappa$-Minkowski noncommutative space time are presented. Real forms of Jordanian deformations are also discussed. The method of similarity transformations is applied to the Poincar\'e-Weyl Hopf algebra and two types of one parameter families of dispersion relations are constructed. Mathematically equivalent deformations, that are related to nonlinear changes of symmetry generators and linked with similarity maps, may lead to differences in the description of physical phenomena.