AI 中文总结
该研究针对李群作用下的集体哈密顿量超可积性提出新方法，利用Guillemin-Sternberg环面作用推导关键等式，补充了相关领域早期研究成果。
AI 中文摘要
我们提出一种针对连通半单紧李群\textit{G}在辛流形\textit{M}上的哈密顿作用下不变的集体哈密顿量超可积性的新方法。通过利用可追溯至Guillemin和Sternberg[GS,1983]的哈密顿环面作用，我们证明了\boldsymbol{\frak{H}:=\boldsymbol{\frak{J}}^*(C^\boldsymbol{\frak{g}}^*)^G}（其中\boldsymbol{\frak{J}: M \to \boldsymbol{\frak{g}}^*}是\textit{G}作用的动量映射）的函数维数，及其在\textit{C}^\boldsymbol{\frak{}}(M)中的中心化子\boldsymbol{\frak{F}}满足等式\boldsymbol{\text{ddim}(\boldsymbol{\frak{H}}) + \text{ddim}(\boldsymbol{\frak{F}}) = \text{dim}(M)}。结合非平凡性条件，这确保了阿贝尔泊松代数\boldsymbol{\frak{H}\boldsymbol{\frak{C}}^\boldsymbol{\frak{}}(M)}构成一个超可积系统，且由此可推出GS环面作用的动量映射给出该系统的作用变量。我们的工作为集体超可积性提供了新的视角，补充了Bolsinov和Jovanović的早期结果。
英文摘要
We present a novel approach to the superintegrability of collective Hamiltonians invariant under a Hamiltonian action of a connected semisimple compact Lie group, $G$, on a symplectic manifold, $M$. By exploiting a Hamiltonian torus action that goes back to Guillemin and Sternberg [GS,1983], we demonstrate that the functional dimensions of $\mathfrak{H} := \mathcal{J}^*(C^\infty(\mathfrak{g}^*)^G)$, where $\mathcal{J}: M \to \mathfrak{g}^*$ is the momentum map of the $G$ action, and its centralizer $\mathfrak{F}$ in $C^\infty(M)$ satisfy the equality $\mathrm{ddim}(\mathfrak{H}) + \mathrm{ddim}(\mathfrak{F}) = \mathrm{dim}(M)$. Together with a non-triviality condition, this ensures that the Abelian Poisson algebra $\mathfrak{H}\subset C^\infty(M)$ represents a superintegrable system, and it also follows that the momentum map of the GS torus action yields action variables for the system. Our work provides a new insight into collective superintegrability complementing earlier results of Bolsinov and Jovanović.
Comments12 pages, v2: corrected the wording of Lemma 2.11 and added a few sentences
