Polynomial Poisson Algebras and Superintegrability on Reductive Homogeneous Spaces

Speaker: Junze Zhang (昆士兰大学)

Time: 2025-12-10 16:30-17:30
Location: 1010, SIMIS
Zoom Meeting ID: 892 8714 0015 Passcode: SIMIS

Abstract:

In this talk, we talk about an algebro-geometric description of constructing superintegrable geodesic and magnetic geodesic flows on homogeneous symplectic manifolds $M=G/A$ of compact semisimple type. In $(T^M,\omega_\varepsilon)$, with $\omega_\varepsilon=\omega_{\mathrm{can}}+\varepsilon\,\pi^\omega_{\mathrm{KKS}}$, we build two canonical and commuting families of polynomial first integrals: One pulled back from the Lie algebra via the moment map, and one pulled back from an $\mathrm{Ad}(A)$-invariant affine slice of $\mathfrak{m}$. Their common image generates a reduced Poisson algebra obtained from a fiber tensor product. In a dense regular locus, the rank of the defining map splits into an orbit part and a fiber part, yielding the identity \begin{align} \mathrm{rank}\,d\Phi=(n-r)+\rho_A, \end{align} where $n=\dim\mathfrak{g}$, $r=\mathrm{rank}\,G$, and $\rho_A=\mathrm{trdeg}\,S(\mathfrak{m}-\varepsilon W)^A$. The resulting projection chain realizes a superintegrable system