Dirac brackets and reduction of invariant bi-Poisson structures

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 $\{(\eta^t)'\}$.