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.
Categories: math.SG math.AG