Hopf algebra gauge theory on a ribbon graph

We generalise the notion of a group gauge theory on a graph embedded into an oriented surface to finite-dimensional ribbon Hopf algebras. By linearising the corresponding structures for groups, we obtain axioms that encode the notions of connections, the algebra of functions on connections, gauge transformations and gauge invariant observables. Together with certain locality conditions, these axioms reduce the construction of a Hopf algebra gauge theory to a basic building block, a Hopf algebra gauge theory for a vertex with n incoming edge ends. The associated algebra of functions is dual to a two-sided twist deformation of the n-fold tensor product of the Hopf algebra. We show that the algebra of functions and the subalgebra of observables for a Hopf algebra gauge theory coincide with the ones obtained in the combinatorial quantisation of Chern-Simons theory, thus providing an axiomatic derivation of the latter. We discuss the notion of holonomy in a Hopf algebra gauge theory and show that for semisimple Hopf algebras this defines a functor from the path groupoid into a certain category associated with the Hopf algebra gauge theory. Curvatures are then obtained as holonomies around the faces of the graph, correspond to central elements of the algebra of observables and define a set of commuting projectors on the subalgebra of observables on flat connections. We show that the algebra of observables and its image under these projectors are topological invariants and depend only on the homeomorphism class of the surface obtained, respectively, by gluing annuli and discs to the faces of the graph