Gelfand-Cetlin abelianizations of symplectic quotients
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909444776919040 |
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| author | Crooks, Peter Weitsman, Jonathan |
| author_facet | Crooks, Peter Weitsman, Jonathan |
| contents | We show that generic symplectic quotients of a Hamiltonian $G$-space $M$ by the action of a compact connected Lie group $G$ are also symplectic quotients of the same manifold $M$ by a compact torus. The torus action in question arises from certain integrable systems on $\mathfrak{g}^*$, the dual of the Lie algebra of $G$. Examples of such integrable systems include the Gelfand-Cetlin systems of Guillemin-Sternberg in the case of unitary and special orthogonal groups, and certain integrable systems constructed for all compact connected Lie groups by Hoffman-Lane. Our abelianization result holds for smooth quotients, and more generally for quotients which are stratified symplectic spaces in the sense of Sjamaar-Lerman. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_04978 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Gelfand-Cetlin abelianizations of symplectic quotients Crooks, Peter Weitsman, Jonathan Symplectic Geometry 53D20 (primary), 17B80 (secondary) We show that generic symplectic quotients of a Hamiltonian $G$-space $M$ by the action of a compact connected Lie group $G$ are also symplectic quotients of the same manifold $M$ by a compact torus. The torus action in question arises from certain integrable systems on $\mathfrak{g}^*$, the dual of the Lie algebra of $G$. Examples of such integrable systems include the Gelfand-Cetlin systems of Guillemin-Sternberg in the case of unitary and special orthogonal groups, and certain integrable systems constructed for all compact connected Lie groups by Hoffman-Lane. Our abelianization result holds for smooth quotients, and more generally for quotients which are stratified symplectic spaces in the sense of Sjamaar-Lerman. |
| title | Gelfand-Cetlin abelianizations of symplectic quotients |
| topic | Symplectic Geometry 53D20 (primary), 17B80 (secondary) |
| url | https://arxiv.org/abs/2209.04978 |