方案理论各向同性还原

Peter Crooks, Maxence Mayrand
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引用次数: 0

摘要

我们开发了一种仿射方案理论版的交映群体哈密顿还原法。它适用于$\Bbbk=\mathbb{R}$或$\Bbbk=\mathbb{C}$,并针对仿交映群元$\mathcal{G}\rightrightarrows X$、仿哈密顿$\mathcal{G}$-scheme$mu:X$, a coisotropic subvariety $S\subseteq X$, and astabilizer subgroupoid $\mathcal{H}\rightrightarrows S$.我们的第一个主要结果是,$\Bbbk[M]$ 上的泊松括号会在子集$\Bbbk[\mu^{-1}(S)]^{\mathcal{H}}$ 上引起泊松括号。然后宣布泊松方案$\mathrm{Spec}(\Bbbk[\mu^{-1}(S)]^{\mathcal{H}})$ 是$M$ 的哈密顿还原。其他主要结果包括$\mathrm{Spec}(\Bbbk[\mu^{-1}(S)]^{\mathcal{H}})$继承残余哈密顿方案结构的充分条件。我们的主要结果最好被视为早先论文的仿射方案理论对应物,在这篇论文中,我们同时归纳了几个哈密顿还原过程。通过这种方式,本研究产生了马斯登-拉蒂乌还原、米卡米-韦恩斯坦还原、尼亚茨基-韦恩斯坦还原以及沿着一般各向异性子满的交点还原的方案理论模拟。这项工作的最初推动力是它对摩尔-立川猜想的广义化和证明的实用性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Scheme-theoretic coisotropic reduction
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 $\mu:M\longrightarrow X$, 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 $\Bbbk[\mu^{-1}(S)]^{\mathcal{H}}$. The Poisson scheme $\mathrm{Spec}(\Bbbk[\mu^{-1}(S)]^{\mathcal{H}})$ is then declared to be a Hamiltonian reduction of $M$. Other main results include sufficient conditions for $\mathrm{Spec}(\Bbbk[\mu^{-1}(S)]^{\mathcal{H}})$ 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, \'{S}niatycki-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.
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