  We prove 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). This shows that Kitaev models are a
special case of the older and more general combinatorial models. This
equivalence is an 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.
  We first prove that the triangle operators of a Kitaev model form a module
algebra over a Hopf algebra of gauge transformations and that this module
algebra is isomorphic to the lattice algebra in the combinatorial formalism.
Both algebras can be viewed as the algebra of functions on gauge fields in a
Hopf algebra gauge theory. The isomorphism between them induces an algebra
isomorphism between their subalgebras of invariants, which are interpreted as
gauge invariant functions or observables. It also relates the curvatures in the
two models, which are given as holonomies around the faces of the lattice. This
yields an isomorphism between the subalgebras obtained by projecting out
curvatures, which can be viewed as the algebras of functions on flat gauge
fields and are the topological invariants of the two models.
