It is sufficient to replace the trivial commutation rules (10) with more general ones.

These commutation rules are not consistent in general, because the Jacobi identities for [mathematical expression not reproducible] are violated.

In fact, the modified commutation rules (13) are not preserved in general by the action (28)-(29).

Even when [[GAMMA].sup.v[lambda].sub.[alpha]] = 0, the deformed commutation rules among [mathematical expression not reproducible] should give rise to relative locality effects under Lorentz transformations.

In the special case of [kappa]-Minkowski, for instance, a popular choice of the coproduct [2] induces nonlinear commutation rules among momenta and boost generators and consequently determines a deformed Casimir operator.

In this respect, we recall that the Hopf structure for the [kappa]-deformation of (2 + 1)D (A)dS and Poincare (P) algebras were collectively obtained in [19], and their connection between their deformed commutation rules and 2 + 1 quantum gravity has been explored in [38].

Next if we denote by [[??].sup.i] the quantum group coordinate dual to [Y.sub.i], such that <[[??].sup.i] | [Y.sub.j]> = [[delta].sup.i.sub.j], and write the cocommutators as [delta]([Y.sub.i]) = [f.sup.jk.sub.i] [Y.sub.j] [conjunction] [Y.sub.k], then Lie bialgebra duality provides the so- called Drinfel'd-double Lie algebra [7, 8] formed by three sets of brackets: the initial Lie algebra, the dual relations, and the crossed commutation rules; namely,

The usual way to propose a noncommutative spacetime is to consider the commutation rules involving the quantum coordinates [[??].sub.[mu]].

The adjoint action on the quantum coordinates [??], [??] of the isotropy subgroup of a worldline spanned by J and [P.sub.0] (6) gives the following nondeformed commutation rules:

Several quantum deformations of nonsemisimple groups have been constructed by applying this procedure; among them we underline the [kappa]-Poincare group [15-18], for which the full set of commutation rules are linear in the deformation parameter.