Scheme-theoretic coisotropic reduction

Abstract

We develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over $\mathbb{k}=\mathbb{R}$ or $\mathbb{k}=\mathbb{C}$, and is formulated for an affine symplectic groupoid $\mathcal{G}\rightrightarrows X$, an affine Hamiltonian $\mathcal{G}$-scheme $\hat{I}¼:M⟶X$, a coisotropic subvariety $S\subseteq X$, and a stabilizer subgroupoid $\mathcal{H}\rightrightarrows S$. Our first main result is that the Poisson bracket on $\mathbb{k}[M]$ induces a Poisson bracket on the subquotient $\mathbb{k}[\hat{I}{¼}^{-1}(S){]}^{\mathcal{H}}$. The Poisson scheme $\mathrm{Spec}(\mathbb{k}[\hat{I}{¼}^{-1}(S){]}^{\mathcal{H}})$ is then declared to be a Hamiltonian reduction of $M$. Other main results include sufficient conditions for $\mathrm{Spec}(\mathbb{k}[\hat{I}{¼}^{-1}(S){]}^{\mathcal{H}})$ 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.