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

Ribbon operators in the generalized Kitaev quantum double model based on Hopf algebras