Hopf-algebra description of noncommutative-space-time symmetries

In the study of certain noncommutative versions of Minkowski space-time a lot remains to be understood for a satisfactory characterization of their symmetries. Adopting as our case study the kappa-Minkowski noncommutative space-time, on which a large literature is already available, we propose a line of analysis of noncommutative-space-time symmetries that relies on the introduction of a Weyl map (connecting a given function in the noncommutative Minkowski with a corresponding function in commutative Minkowski). We provide new elements in favor of the expectation that the commutative-space-time notion of Lie-algebra symmetries must be replaced, in the noncommutative-space-time context, by the one of Hopf-algebra symmetries. While previous studies appeared to establish a rather large ambiguity in the description of the Hopf-algebra symmetries of kappa-Minkowski, the approach here adopted reduces the ambiguity to the description of the translation generators, and our results, independently of this ambiguity, are sufficient to clarify that some recent studies which argued for an operational indistinguishability between theories with and without a length-scale relativistic invariant, implicitly assumed that the underlying space-time would be classical. Moreover, while usually one describes theories in kappa-Minkowski directly at the level of equations of motion, we explore the nature of Hopf-algebra symmetry transformations on an action.