  Kitaev's quantum double model is a family of exactly solvable lattice models
that realize two dimensional topological phases of matter. Originally it is
based on finite groups, and is later generalized to semi-simple Hopf algebras.
We rigorously define and study ribbon operators in the generalized Kitaev
quantum double model. These ribbon operators are important tools to understand
quasi-particle excitations. It turns out that there are some subtleties in
defining the operators in contrast to what one would naively think. In
particular, one has to distinguish two classes of ribbons which we call locally
clockwise and locally counterclockwise ribbons. Moreover, this issue already
exists in the original model based on finite non-Abelian groups. We show how
certain properties would fail even in the original model if we do not
distinguish these two classes of ribbons. Perhaps not surprisingly, under the
new definitions ribbon operators satisfy all properties that are expected. For
instance, they create quasi-particle excitations only at the end of the ribbon,
and the types of the quasi-particles correspond to irreducible representations
of the Drinfeld double of the input Hopf algebra. However, the proofs of these
properties are much more complicated than those in the case of finite groups.
This is partly due to the complications in dealing with general Hopf algebras
rather than just group algebras.
