A cyclic integral on κ-Minkowski noncommutative space-time

We examine some alternative possibilities for an action functional for kappa-Minkowski noncommutative space-time. Early works on kappa-Minkowski focused on kappa-Poincare covariance and the dependence of the action functional on the choice of Weyl map, renouncing to invariance under cyclic permutations of the factors composing the argument of the action functional. It has been recently suggested, focusing on a specific choice of Weyl map and setting aside the issue of kappa-Poincare covariance of the action functional, that a cyclicity-inducing measure could be introduced in implicit form. We provide an explicit formula for (and derivation of) a choice of measure which indeed ensures cyclicity of the action functional. The same choice of measure is applicable to all the most used choices of Weyl map, but we find that this cyclicity-inducing measure is not invariant under kappa-Poincare transformations. We also notice that the cyclicity-inducing measure can be straightforwardly derived using a map which connects the kappa-Minkowski space-time coordinates and the space-time coordinates of a canonical noncommutative space-time, with coordinate-independent commutators.