  We study the quantum mechanics of a system of topologically interacting
particles in 2+1 dimensions, which is described by coupling the particles to a
Chern-Simons gauge field of an inhomogeneous group. Analysis of the phase space
shows that for the particular case of ISO(3) Chern-Simons theory the underlying
symmetry is that of the quantum double D(SU(2)), based on the homogeneous part
of the gauge group. This in contrast to the usual q-deformed gauge group
itself, which occurs in the case of a homogeneous gauge group. Subsequently, we
describe the structure of the quantum double of a continuous group and the
classification of its unitary irreducible representations. The comultiplication
and the R-element of the quantum double allow for a natural description of the
fusion properties and the nonabelian braid statistics of the particles. These
typically manifest themselves in generalised Aharonov-Bohm scattering
processes, for which we compute the differential cross sections. Finally, we
briefly describe the structure of D(SO(2,1)), the underlying quantum double
symmetry of (2+1)-dimensional quantum gravity.
