Flat bi-Hamiltonian structures and invariant densities

Abstract

A bi-Hamiltonian structure is a pair of Poisson structures $\mathcal{P}$, $\mathcal{Q}$ which are compatible, meaning that any linear combination $\alpha \mathcal{P}+\beta \mathcal{Q}$ is again a Poisson structure. A bi-Hamiltonian structure $(\mathcal{P},\mathcal{Q})$ is called flat if $\mathcal{P}$ and $\mathcal{Q}$ can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic bi-Hamiltonian structure $(\mathcal{P},\mathcal{Q})$ on an odd-dimensional manifold is flat if and only if there exists a local density which is preserved by all vector fields Hamiltonian with respect to $\mathcal{P}$, as well as by all vector fields Hamiltonian with respect to $\mathcal{Q}$.

10 pages; Lett Math Phys (2016)

References in corpus (1)

• Curvature of Poisson pencils in dimension three