Poisson Centralisers and Polynomial Superintegrability for Magnetic Geodesic Flows on Reductive Homogeneous Spaces

Abstract

We provide a method for formulating superintegrable magnetic geodesic flows on reductive homogeneous spaces $M=G/A$, with $G$ a compact semisimple Lie group and $A$ a closed subgroup of $G$. In the twisted cotangent bundle $(T^{\ast }M,\ddot{I}{‰}_{\epsilon })$, with $\ddot{I}{‰}_{\epsilon }=\ddot{I}{‰}_{\mathrm{can}}+\epsilon \ddot{I}{€}^{\ast }\ddot{I}{‰}_{\mathrm{KKS}}$ being the canonical plus Kirillov-Kostant-Souriau (KKS) forms, we build two canonical and commuting families of polynomial first integrals: one pulled back from the Lie algebra $\mathfrak{g}$ of $G$ via the magnetic moment map $P$, and one pulled back from a $\mathrm{Ad}(A)$-invariant affine slice of $\mathfrak{m}\cong T_{eA}M$, where $eA$ is the identity of $G/A$. Their common image generates a reduced Poisson algebra obtained from a fiber tensor product, and the natural multiplication map into a Poisson subalgebra of polynomial functions $\mathcal{O}(T^{\ast }M)\subset C^{\infty }(T^{\ast }M)$ is Poisson and injective. The center of this fiber tensor product is contained in the Poisson center of the symmetric algebra of $\mathfrak{g}$. In a dense regular locus, the resulting projection chain realises a superintegrable system. As examples, two $\mathrm{SU}(3)$ cases are studied (regular torus and irregular $\mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(1))$ quotients), which illustrate the construction and produce explicit action-angle coordinates.