The classical dynamic symmetry for the Sp(1)-Kepler problems

A Poisson realization of the simple real Lie algebra so*(4n) on the phase space of each Sp(1)-Kepler problem is exhibited. As a consequence, one obtains the Laplace-Runge-Lenz vector for each classical Sp(1)-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea of Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of so*(4n) on the Poisson manifold T*H*n/Sp(1). (Here H*n≔Hn\{0} and the Hamiltonian action of Sp(1) on T*H*n is induced from the natural right action of Sp(1) on H*n.)