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We prove that Kitaev's lattice model for a finite-dimensional semisimple Hopfalgebra H is equivalent to the combinatorial quantisation of Chern-Simonstheory for the Drinfeld double D(H). This shows that Kitaev models are aspecial case of the older and more general combinatorial models. Thisequivalence is an analogue of the relation between Turaev-Viro andReshetikhin-Turaev TQFTs and relates them to the quantisation of moduli spacesof flat connections.  We show that the topological invariants of the two models, the algebra ofoperators acting on the protected space of the Kitaev model and the quantummoduli algebra from the combinatorial quantisation formalism, are isomorphic.This is established in a gauge theoretical picture, in which both models appearas Hopf algebra valued lattice gauge theories.  We first prove that the triangle operators of a Kitaev model form a modulealgebra over a Hopf algebra of gauge transformations and that this modulealgebra is isomorphic to the lattice algebra in the combinatorial formalism.Both algebras can be viewed as the algebra of functions on gauge fields in aHopf algebra gauge theory. The isomorphism between them induces an algebraisomorphism between their subalgebras of invariants, which are interpreted asgauge invariant functions or observables. It also relates the curvatures in thetwo models, which are given as holonomies around the faces of the lattice. Thisyields an isomorphism between the subalgebras obtained by projecting outcurvatures, which can be viewed as the algebras of functions on flat gaugefields and are the topological invariants of the two models.

Kitaev lattice models as a Hopf algebra gauge theory