The Poisson Realization of so(2, 2k+2) on Magnetic Leaves

Let ${\mathbb{R}}_{\ast }^{2k+1}={\mathbb{R}}^{2k+1}\setminus \{\vec{0}\}$ ($k\ge 1$) and $\pi$: ${\mathbb{R}}_{\ast }^{2k+1}\to {\mathrm{S}}^{2k}$ be the map sending $\vec r\in {\mathbb R}^{2k+1}_*$to${\vec r\over |\vec r|}\in $\mathrm{S}$^{2k}$. Denote by $P\to {\mathbb{R}}_{\ast }^{2k+1}$ the pullback by $\pi$ of the canonical principal $\mathrm{S}\mathrm{O}(2k)$-bundle $\mathrm{S}\mathrm{O}(2k+1)\to {\mathrm{S}}^{2k}$. Let $E_{\mathrm{♯}}\to {\mathbb{R}}_{\ast }^{2k+1}$ be the associated co-adjoint bundle and $E^{\mathrm{♯}}\to T^{\ast }{\mathbb{R}}_{\ast }^{2k+1}$ be the pullback bundle under projection map $T^{\ast }{\mathbb{R}}_{\ast }^{2k+1}\to {\mathbb{R}}_{\ast }^{2k+1}$. The canonical connection on $\mathrm{S}\mathrm{O}(2k+1)\to {\mathrm{S}}^{2k}$ turns $E^{\mathrm{♯}}$ into a Poisson manifold. The main result here is that the real Lie algebra $\mathfrak{s}\mathfrak{o}(2,2k+2)$ can be realized as a Lie subalgebra of the Poisson algebra $(C^\infty(\mathcal O^\sharp), {, })$,where$\mathcal O^\sharp$isasymplecticleaveof$E^\sharp$ 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.