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.

We study the noncommutative geometry of the Moyal plane from a metric point of view. Starting from a non compact spectral triple based on the Moyal deformation A of the algebra of Schwartz functions on R 2 , we explicitly compute Connes' spectral distance between the pure states of A corresponding to eigenfunctions of the quantum harmonic oscillator. For other pure states, we provide a lower bound to the spectral distance, and show that the latest is not always finite. As a consequence, we show that the spectral triple [19] is not a spectral metric space in the sense of [5]. This motivates the study of truncations of the spectral triple, based on M n (C) with arbitrary n ∈ N, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n = 2.

Azienda Ospedaliera Universitaria Maggiore della Carità.

We study the spectral geometry of the quantum projective plane CP 2 q , a deformation of the complex projective plane CP 2 , the simplest example of a spin c manifold which is not spin. In particular, we construct a Dirac operator D which gives a 0 + -summable spectral triple, equivariant under U q (su(3)). The square of D is a central element for which left and right actions on spinors coincide, a fact that is exploited to compute explicitly its spectrum. MSC (2000): 58B34, 17B37.

The realization that forthcoming experimental studies, such as the ones planned for the GLAST space telescope, will be sensitive to Planck-scale deviations from Lorentz symmetry has increased interest in noncommutative spacetimes in which this type of effects is expected. We focus here on κ-Minkowski spacetime, a muchstudied example of Lie-algebra noncommutative spacetime, but our analysis appears to be applicable to a more general class of noncommutative spacetimes. A technical controversy which has significant implications for experimental testability is the one concerning the κ-Minkowski relation between group velocity and momentum. A large majority of studies adopted the relation v = dE(p)/dp, where E(p) is the κ-Minkowski dispersion relation, but recently some authors advocated alternative formulas. While in these previous studies the relation between group velocity and momentum was introduced through ad hoc formulas, we rely on a direct analysis of wave propagation in κ-Minkowski. Our results lead conclusively to the relation v = dE(p)/dp. We also show that the previous proposals of alternative velocity/momentum relations implicitly relied on an inconsistent implementation of functional calculus on κ-Minkowski and/or on an inconsistent description of spacetime translations.