Topological Quantum Groups, Star Products and their relations

. This short summary of recent developments in quantum compact groups and star products is divided into 2 parts. In the first one we recast star products in a more abstract form as deformations and review its recent developments. The second part starts with a rapid presentation of standard quantum group theory and its problems, then moves to their completion by introduction of suitable Montel topologies well adapted to duality. Preferred deformations (by star products and unchanged coproducts) of Hopf algebras of functions on compact groups and their duals, are of special interest. Connection with the usual models of quantum groups and the quantum double is then presented. 0 - INTRODUCTION. The idea that quantum theories are deformations of classical theories was presumably in the back of the mind of many scientists, even before the mathematical notion of deformation was formalized by Gerstenhaber [G] for algebraic structures. We were even told by witnesses (many of whom contribute..