Over these past few years several quantum-gravity research groups have been exploring the possibility that in some Planck-scale nonclassical descriptions of spacetime one or another form of nonclassical spacetime symmetries might arise. One of the most studied scenarios is based on the use of Hopf algebras, but previous attempts were not successful in deriving constructively the properties of the conserved charges one would like to obtain from the Hopf structure, and this in turn did not allow a crisp physical characterization of the new concept of spacetime symmetry. Working within the example of κ-Minkowski noncommutative spacetime, known to be particularly troublesome from this perspective, we observe that these past failures in the search of the charges originated from not recognizing the crucial role that the noncommutative differential calculus plays in the symmetry analysis. We show that, if the properties of the κ-Minkowski differential calculus are correctly taken into account, one can easily perform all the steps of the Noether analysis and obtain an explicit formula relating fields and energy-momentum charges. Our derivation also exposes the fact that an apparent source of physical ambiguity in the description of the Hopf-algebra rules of action, which was much emphasized in the literature, actually only amounts to a choice of conventions and in particular does not affect the formulas for the charges.1 The space indices j, l take values in {1, 2, 3} while 0 is the time index. We shall later also use the spacetime indices µ, ν, α, which take values in {0, 1, 2, 3}.2 Rather than our length scale λ a majority of authors use the energy scale κ, which is the inverse of λ (λ → 1/κ). 3 One notices that κ-Minkowski and the κ-Poincaré Hopf algebra form a "Heisenberg double" [8,9], i.e. κ-Minkowski and κ-Poincaré are linked, as algebras, in a way that is rather similar to the relationship between classical Minkowski spacetime and the classical Poincaré Lie algebra.

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 κ-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 κ-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 κ-Minkowski directly at the level of equations of motion, we explore the nature of Hopf-algebra symmetry transformations on an action.

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We construct a Dirac equation that is consistent with one of the recently-proposed schemes for a "doubly-special relativity", a relativity with both an observer-independent velocity scale (still naturally identified with the speed-of-light constant) and an observer-independent length/momentum scale (possibly given by the Planck length/momentum). We find that the introduction of the second observer-independent scale only induces a mild deformation of the structure of Dirac spinors. We also show that our modified Dirac equation naturally arises in constructing a Dirac equation in the κ-Minkowski noncommutative spacetime. Previous, more heuristic, studies had already argued for a possible role of doubly-special relativity in κ-Minkowski, but remained vague on the nature of the consistency requirements that should be implemented in order to assure the observer-independence of the two scales. We find that a key role is played by the choice of a differential calculus in κ-Minkowski. A much-studied choice of the differential calculus does lead to our doublyspecial relativity Dirac equation, but a different scenario is encountered for another popular choice of differential calculus.

We introduce the notion of generalized Weyl system, and use it to define * -products which generalize the commutation relations of Lie algebras. In particular we study in a comparative way various * -products which generalize the κ-Minkowski commutation relation.