THE POISSON REALIZATION OF so(2, 2k + 2) ON MAGNETIC LEAVES

Let R ∗ = R2k+1 \ {~0} (k ≥ 1) and π: R ∗ → S2k be the map sending ~r ∈ R ∗ to ~r |~r| ∈ S 2k . Denote by P → R ∗ the pullback by π of the canonical principal SO(2k)-bundle SO(2k + 1) → S2k . Let E] → R ∗ be the associated co-adjoint bundle and E] → T ∗R ∗ be the pullback bundle under projection map T ∗R ∗ → R ∗ . The canonical connection on SO(2k + 1) → S2k turns E] into a Poisson manifold. The main result here is that the real Lie algebra so(2, 2k + 2) can be realized as a Lie subalgebra of the Poisson algebra (C∞(O]), {, }), whereO] is a symplectic leave of E] of special kind. Consequently, in view of the earlier result of the author, an extension of the classical MICZ Kepler problems to dimension 2k + 1 is obtained. The hamiltonian, the angular momentum, the Lenz vector and the equation of motion for this extension are all explicitly worked out. CONTENTS