  We introduce $*$-structures on braided groups and braided matrices. Using
this, we show that the quantum double $D(U_q(su_2))$ 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 $U_q(su_2)$. 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 $U_q(su_2^*)$ where $su_2^*$ is the
Drinfeld dual Lie algebra of $su_2$. Similar results hold for the quantum
double and its dual of a general quantum group.
