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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.11932 |
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| _version_ | 1866914258953961472 |
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| author | Crooks, Peter Mayrand, Maxence |
| author_facet | Crooks, Peter Mayrand, Maxence |
| contents | We develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over $\Bbbk=\mathbb{R}$ or $\Bbbk=\mathbb{C}$, and is formulated for an affine symplectic groupoid $\mathcal{G}\rightrightarrows X$, an affine Hamiltonian $\mathcal{G}$-scheme $μ:M\longrightarrow X$, a coisotropic subvariety $S\subseteq X$, and a stabilizer subgroupoid $\mathcal{H}\rightrightarrows S$. Our first main result is that the Poisson bracket on $\Bbbk[M]$ induces a Poisson bracket on the subquotient $\Bbbk[μ^{-1}(S)]^{\mathcal{H}}$. The Poisson scheme $\mathrm{Spec}(\Bbbk[μ^{-1}(S)]^{\mathcal{H}})$ is then declared to be a Hamiltonian reduction of $M$. Other main results include sufficient conditions for $\mathrm{Spec}(\Bbbk[μ^{-1}(S)]^{\mathcal{H}})$ to inherit a residual Hamiltonian scheme structure.
Our main results are best viewed as affine scheme-theoretic counterparts to an earlier paper, where we simultaneously generalize several Hamiltonian reduction processes. In this way, the present work yields scheme-theoretic analogues of Marsden-Ratiu reduction, Mikami-Weinstein reduction, Śniatycki-Weinstein reduction, and symplectic reduction along general coisotropic submanifolds. The initial impetus for this work was its utility in formulating and proving generalizations of the Moore-Tachikawa conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11932 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scheme-theoretic coisotropic reduction Crooks, Peter Mayrand, Maxence Symplectic Geometry Algebraic Geometry 17B63 (primary), 14L30, 53D17 (secondary) We develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over $\Bbbk=\mathbb{R}$ or $\Bbbk=\mathbb{C}$, and is formulated for an affine symplectic groupoid $\mathcal{G}\rightrightarrows X$, an affine Hamiltonian $\mathcal{G}$-scheme $μ:M\longrightarrow X$, a coisotropic subvariety $S\subseteq X$, and a stabilizer subgroupoid $\mathcal{H}\rightrightarrows S$. Our first main result is that the Poisson bracket on $\Bbbk[M]$ induces a Poisson bracket on the subquotient $\Bbbk[μ^{-1}(S)]^{\mathcal{H}}$. The Poisson scheme $\mathrm{Spec}(\Bbbk[μ^{-1}(S)]^{\mathcal{H}})$ is then declared to be a Hamiltonian reduction of $M$. Other main results include sufficient conditions for $\mathrm{Spec}(\Bbbk[μ^{-1}(S)]^{\mathcal{H}})$ to inherit a residual Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to an earlier paper, where we simultaneously generalize several Hamiltonian reduction processes. In this way, the present work yields scheme-theoretic analogues of Marsden-Ratiu reduction, Mikami-Weinstein reduction, Śniatycki-Weinstein reduction, and symplectic reduction along general coisotropic submanifolds. The initial impetus for this work was its utility in formulating and proving generalizations of the Moore-Tachikawa conjecture. |
| title | Scheme-theoretic coisotropic reduction |
| topic | Symplectic Geometry Algebraic Geometry 17B63 (primary), 14L30, 53D17 (secondary) |
| url | https://arxiv.org/abs/2408.11932 |