Dirac brackets and reduction of invariant bi-Poisson structures

TL;DR

This paper studies how bi-Poisson structures on manifolds with group actions can be reduced to submanifolds associated with isotropy groups, simplifying their analysis through invariant functions and group actions.

Contribution

It proves that the bi-Poisson structure on the original manifold induces a related structure on a submanifold with a proper, locally free group action, facilitating reduction.

Findings

The induced bi-Poisson structure on the submanifold is invariant under a quotient group.

Invariant functions on the original manifold correspond to those on the submanifold.

Reduction simplifies the study of bi-Poisson structures via group action analysis.

Abstract

Let $X$ be a manifold with a bi-Poisson structure $\{\eta^t\}$ generated by a pair of $G$-invariant symplectic structures $\omega_1$ and $\omega_2$, where the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\mathfrak{h}$ be the submanifold of $X$ consisting of the points in $X$ with the stabilizer algebra equal to the Lie algebra $\mathfrak{h}$ of $H$ and with the stabilizer group conjugated to $H$ in $G$. We prove that the pair of symplectic structures $\omega_1|_{X^r_\mathfrak{h}}$ and $\omega_2|_{X^r_\mathfrak{h}}$ generates an $N(H^0)/H^0$-invariant bi-Poisson structure on $X^r_\mathfrak{h}$, where $N(H^0)$ is the normalizer in $G$ of the identity component $H^0$ of $H$. The action of $\widetilde G=N(H^0)/H^0$ on $X^r_\mathfrak{h}$ is locally free and proper and, moreover, the spaces $A^G$ of $G$-invariant functions on $X$ and $A^{\widetilde G}$ of $\widetilde G$-invariant functions on $X^r_\mathfrak{h}$ can be canonically identified and therefore the bi-Poisson structure $\{(\eta^t)'\}$ induced on $A^G\simeq A^{\widetilde G}$ can be treated as the reduction with respect to a {\em locally free} action of a Lie group which essentially simplifies the study of $\{(\eta^t)'\}$.