On the spectral and homological dimension of k-Minkowski space

We extend the construction of a spectral triple for k-Minkowski space, previously given for the two-dimensional case, to the general n-dimensional case. This takes into account the modular group naturally arising from the symmetries of the geometry, and requires the use of notions that have been recently developed in the frameworks of twisted and modular spectral triples. First we compute the spectral dimension, using an appropriate weight, and show that in general it coincides with the classical one. We also study the classical limit and the analytic continuation of the associated zeta function. Then we compare this notion of dimension with the one coming from homology. To this end, we compute the twisted Hochschild dimension of the universal enveloping algebra underlying k-Minkowski space. The result is that twisting avoids the dimension drop, similarly to other examples coming from quantum groups. In particular, the simplest such twist is given by the inverse of the modular group mentioned above.