Geometric construction of superintegrable Poisson projection chains via Poisson centralizers

TL;DR

This paper develops a geometric method to construct superintegrable systems using Poisson centralizers within Lie-Poisson algebras, linking algebraic structures to integrability properties.

Contribution

It introduces a framework based on Poisson projection chains derived from reductive subgroup chains, providing explicit constructions of superintegrable systems.

Findings

Identifies superintegrable chains via inclusions of invariant Poisson subalgebras.

Determines quotient maps that organize the algebraic structure of integrable systems.

Provides examples illustrating the organization of centralizer generators into superintegrable chains.

Abstract

We introduce a geometric framework for constructing superintegrable systems from Poisson centralizers (commutants) in the Lie-Poisson algebra $S(\mathfrak{g})$ of a complex semisimple Lie algebra. Starting from a chain of reductive subgroups, we study the corresponding invariant Poisson subalgebras and their Poisson centers, and formulate superintegrability in terms of a \emph{Poisson projection chain} of affine Poisson varieties. For a maximal torus $T\subset G$, we prove that the inclusions $S(\mathfrak{g})^G\subset S(\mathfrak{g})^T\subset S(\mathfrak{g})$ determine a superintegrable chain and identify the associated quotient maps $\mathfrak{g}\xrightarrow{\chi_T}\mathfrak{g}//T\xrightarrow{\rho}\mathfrak{g}//G$. The rank (transcendence degree) computations yield the expected dimension split between commuting Hamiltonians and first integrals, and we describe the corresponding symplectic leaves in the intermediate space. Several examples illustrate how the centralizer generators organize into explicit superintegrable Poisson chains.