  We construct global observable algebras and global DHR morphisms for the
Virasoro minimal models with central charge c(2,q), q odd. To this end, we pass
{}from the irreducible highest weight modules to path representations, which
involve fusion graphs of the c(2,q) models. The paths have an interpretation in
terms of quasi-particles which capture some structure of non-conformal
perturbations of the c(2,q) models. The path algebras associated to the path
spaces serve as algebras of bounded observables. Global morphisms which
implement the superselection sectors are constructed using quantum symmetries:
We argue that there is a canonical semi-simple quantum symmetry algebra for
each quasi-rational CFT, in particular for the c(2,q) models. These symmetry
algebras act naturally on the path spaces, which allows to define a global
field algebra and covariant multiplets therein.
