TOPOLOGICAL QUANTUM DOUBLE

Following a preceding paper showing how the introduction of a t.v.s. t opology on quantum groups led to a remarkable unification and rigidifi cation of the different definitions, we adapt here, in the same way, t he definition of quantum double. This topological double is dualizable and reflexive (even for infinite dimensional algebras). In a simple c ase we show, considering the double as the ''zero class'' of an extens ion theory, the uniqueness of the double structure as a quasi-Hopf alg ebra.