Hopf algebra gauge theory on a ribbon graph

We generalise gauge theory on a graph so that the group becomes finite-dimensional ribbon Hopf algebra, graph, and gauge-theoretic concepts such as connections, transformations observables are replaced by linearised analogues. Starting from physical considerations, we derive an axiomatic definition of Hopf-algebra theory, including locality conditions under which for general can be assembled local data in neighbourhood each vertex. For vertex with n incoming edge ends, algebra non-commutative "functions" connections is dual to two-sided twist deformation n-fold tensor power algebra. show these algebras assemble give functions gauge-invariant subalgebra "observables" coincide those obtained combinatorial quantisation Chern-Simons thus providing derivation latter. then discuss holonomy semisimple this gives, path embedded map into depending functorially path. Curvatures -- holonomies around faces canonically associated correspond central elements observables, define set commuting projectors onto flat connections. The all or topological invariants, only topology, respectively, punctured closed surface gluing annuli discs along edges graph.