Geometric construction of superintegrable Poisson projection chains via Poisson centralizers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911684426203136 |
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| author | Jiang, Kai Ma, Guorui Marquette, Ian Zhang, Junze Zhang, Yao-Zhong |
| author_facet | Jiang, Kai Ma, Guorui Marquette, Ian Zhang, Junze Zhang, Yao-Zhong |
| contents | 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 $\mathfrak{g}\xrightarrow{χ_T}\mathfrak{g}//T\xrightarrowρ\mathfrak{g}//G$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_14490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric construction of superintegrable Poisson projection chains via Poisson centralizers Jiang, Kai Ma, Guorui Marquette, Ian Zhang, Junze Zhang, Yao-Zhong Mathematical Physics 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 $\mathfrak{g}\xrightarrow{χ_T}\mathfrak{g}//T\xrightarrowρ\mathfrak{g}//G$. 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. |
| title | Geometric construction of superintegrable Poisson projection chains via Poisson centralizers |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2605.14490 |