We show that Kitaev's lattice model for a finite-dimensional semisimple
Hopf algebra H is equivalent to the combinatorial quantisation of
Chern-Simons theory for the Drinfeld double D(H). As a result, Kitaev
models are a special case of combinatorial quantization of Chern-Simons
theory by Buffenoir and Roche and Alekseev, Grosse and Schomerus.
This equivalence is a Hamiltonian analogue of the relation between
Turaev-Viro and Reshetikhin-Turaev TQFTs and relates them to the
quantisation of moduli spaces of flat connections.
We show that the topological invariants of the two models, the algebra
of operators acting on the protected space of the Kitaev model and the
quantum moduli algebra from the combinatorial quantisation formalism,
are isomorphic. This is established in a gauge theoretical picture, in
which both models appear as Hopf algebra valued lattice gauge theories.