  We revisit the notion of quantum Lie algebra of symmetries of a
noncommutative spacetime, its elements are shown to be the generators of
infinitesimal transformations and are naturally identified with physical
observables. Wave equations on noncommutative spaces are derived from a quantum
Hodge star operator. This general noncommutative geometry construction is then
exemplified in the case of k-Minkowski spacetime. The corresponding quantum
Poincare'-Weyl Lie algebra of infinitesimal translations, rotations and
dilatations is obtained. The d'Alembert wave operator coincides with the
quadratic Casimir of quantum translations and it is deformed as in Deformed
Special Relativity theories. Also momenta (infinitesimal quantum translations)
are deformed, and correspondingly the Einstein-Planck relation and the de
Broglie one. The energy-momentum relations (dispersion relations) are
consequently deduced. These results complement those of the phenomenological
literature on the subject.
