The q-Euclidean algebra U_q(e^N) and the corresponding q-Euclidean lattice

We present the Euclidean Hopf algebra $U_q(e^N)$ dual of $Fun(R_q^N\lcross SO_{q^{-1}}(N))$ and describe its fundamental Hilbert space representations \cite{fioeu}, which turn out to be rather simple `` lattice-regularized '' versions of the classical ones, in the sense that the spectra of squared momentum components are discrete and the corresponding eigenfunctions normalizable.These representations can be regarded as describing a quantum system consisting of one free particle on the quantum Euclidean space. A suitable notion of classical limit is introduced, so that we recover the classical continuous spectra and generalized (non-normalizable) eigenfunctions in that limit.