THE QUANTUM DOUBLE AS QUANTUM-MECHANICS

We introduce -structures on braided groups and braided matrices. Usin g this, we show that the quantum double D(U(q)(su2)) can be viewed as the quantum algebra of observables of a quantum particle moving on a h yperboloid in q-Minkowski space (a three-sphere in the Lorentz metric) , and with the role of angular momentum played by U(q)(su2). This prov ides a new example of a quantum system whose algebra of observables is a Hopf algebra. Furthermore, its dual Hopf algebra can also be viewed as a quantum algebra of observables, of another quantum system. This time the position space is a q-deformation of SL (2, R) and the moment um group is U(q) (su2) where su2* is the Drinfeld dual Lie algebra of su2. Similar results hold for the quantum double and it-s dual of a g eneral quantum group.