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

Abstract

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

14 pages

References in corpus (2)

• Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems
• Lorentz Group and Oriented MICZ-Kepler Orbits

Cited by in corpus (8)

• The Universal Kepler Problem
• MICZ Kepler Systems in Noncommutative Space and Duality of Force Laws
• Tulczyjew's Approach for Particles in Gauge Fields
• Particle Motion in Monopoles and Geodesics on Cones
• The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems
• The classical dynamic symmetry for the $\mathrm{U}(1)$-Kepler problems
• Particle Motion in Generalized Dirac's Monopoles of dimension 2k+1
• Hamilton-Dirac systems for charged particles in gauge fields