TOPOLOGICAL FIELD-THEORY AND THE QUANTUM DOUBLE OF SU(2)

We study the quantum mechanics of a system of topologically interactin g particles in 2 + 1 dimensions, which is described by coupling the pa rticles to a Chern-Simons gauge field of an inhomogeneous group. Analy sis of the phase space shows that for the particular case of ISO(3) Ch ern-Simons theory the underlying symmetry is that of the quantum doubl e 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 t he case of a homogeneous gauge group. Subsequently, we describe the st ructure of the quantum double of a continuous group and the classifica tion of its unitary irreducible representations. The comultiplication and the R-element of the quantum double allow for a natural descriptio n of the fusion properties and the non-abelian braid statistics of the particles. These typically manifest themselves in generalised Aharono v-Bohm scattering processes, for which we compute the differential cro ss sections. Finally, we briefly describe the structure of D(SO(2, 1)) , the underlying quantum double symmetry of (2 + 1)-dimensional quantu m gravity. (C) 1998 Elsevier Science B.V.