The Quantum Double as Quantum Mechanics

We introduce ∗-structures on braided groups and braided matrices. Using this, we show that the quantum double D(Uq(su2)) can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in q-Minkowski space (a three-sphere in the Lorentz metric), and with the role of angular momentum played by Uq(su2). This provides 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 momentum group is Uq(su ∗ 2) where su2 is the Drinfeld dual Lie algebra of su2. Similar results hold for the quantum double and its dual of a general quantum group.