  We propose a nonstandard approach to solving the apparent incompatibility
between the coalgebra structure of some inhomogeneous quantum groups and their
natural complex conjugation. In this work we sketch the general idea and
develop the method in detail on a toy-model; the latter is a q-deformation of
the Hopf algebra of 1-dim translations + dilatations. We show how to get all
Hilbert space representations of the latter from tensor products of the
fundamental ones; physically, this corresponds to constructing composite
systems of many free distinct q-particles in terms of the basic one-particle
ones. The spectrum of the total momentum turns out to be the same as that of a
one-particle momentum, i.e. of the form $\{\mu q^n\}_{n\in\zn}$.
