  Two one-parameter families of twists providing kappa-Minkowski * -product
deformed spacetime are considered: Abelian and Jordanian. We compare the
derivation of quantum Minkowski space from two perspectives. The first one is
the Hopf module algebra point of view, which is strictly related with
Drinfeld's twisting tensor technique. The other one relies on an appropriate
extension of "deformed realizations" of nondeformed Lorentz algebra by the
quantum Minkowski algebra. This extension turns out to be de Sitter Lie
algebra. We show the way both approaches are related. The second path allows us
to calculate deformed dispersion relations for toy models ensuing from
different twist parameters. In the Abelian case one recovers kappa-Poincar'e
dispersion relations having numerous applications in doubly special relativity.
Jordanian twists provide a new type of dispersion relations which in the
minimal case (related to Weyl-Poincar'e algebra) takes an energy-dependent
linear mass deformation form.
