## Abstract

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→CP^{n−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 CP^{n−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.

Original language | English (US) |
---|---|

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Journal of Geometry and Physics |

Volume | 124 |

DOIs | |

State | Published - Jan 2018 |

## Keywords

- Dynamic symmetry
- Jordan algebra
- Kepler problem
- Laplace–Runge–Lenz vector

## ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology