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

For the Jordan algebra of hermitian matrices of order n≥2, we let X be its submanifold consisting of rank-one semi-positive definite elements. The composition of the cotangent bundle map π_X: T^∗X→X with the canonical map X→CPⁿ⁻¹ (i.e., the map that sends a given hermitian matrix to its column space), pulls back the Kähler form of the Fubini–Study metric on CPⁿ⁻¹ to a real closed differential two-form ω_K on T^∗X. Let ω_X be the canonical symplectic form on T^∗X and μ a real number. A standard fact says that ω_μ≔ω_X+2μω_K turns T^∗X into a symplectic manifold, hence a Poisson manifold with Poisson bracket {,}_μ. In this article we exhibit a Poisson realization of the simple real Lie algebra su(n,n) on the Poisson manifold (T^∗X,{,}_μ), i.e., a Lie algebra homomorphism from su(n,n) to C^∞(T^∗X,R),{,}_μ. Consequently one obtains the Laplace–Runge–Lenz vector for the classical U(1)-Kepler problem of level n and magnetic charge μ. Since the McIntosh–Cisneros–Zwanziger–Kepler problems (MICZ-Kepler Problems) are the U(1)-Kepler problems of level 2, the work presented here is a direct generalization of the work by A. Barut and G. Bornzin (1971) on the classical dynamic symmetry for the MICZ-Kepler problems.