Narrow Results

An analogue of the Hofer metric ${\varrho }_H$ on the Hamiltonian group $\mathrm{\text{Ham}}(M,\mathrm{\Lambda })$ of a Poisson manifold $(M,\mathrm{\Lambda })$ can be defined, but there is the problem of its nondegeneracy. First, we observe that ${\varrho }_H$ is a genuine metric on $\mathrm{\text{Ham}}(M,\mathrm{\Lambda })$, when the union of all proper leaves of the corresponding symplectic foliation is dense. Next, we deal with the important class of integrable Poisson manifolds. Recall that a Poisson manifold is called integrable, if it can be realized as the space of units of a symplectic groupoid. Our main result states that ${\varrho }_H$ is a Hofer type metric for every Poisson manifold, which admits a Hausdorff integration.

The purpose of this paper is to study multi-derivations of Poisson algebras. First, we show that the space of multi-derivations of a Lie algebra $𝔤$ with the Nijenhuis–Richardson bracket is a differential graded Lie algebra. Then we introduce the notion of a multi-derivation of a Poisson algebra, and show that the space of multi-derivations of a Poisson algebra with the Nijenhuis–Richardson bracket is also a graded Lie algebra. Finally, we study multi-derivations of three concrete Poisson algebras coming from linear Poisson manifolds, symplectic manifolds and Poisson manifolds with vanishing first cohomology groups. We establish the relationship between multi-derivations of a Lie algebra $𝔤$ and multi-derivations of the linear Poisson manifold ${𝔤}^{\ast }$, and show that a two-derivation of a connected symplectic manifold $(M,\omega )$ is $\lambda \pi$, where $\pi$ is the Poisson bivector field induced by the symplectic structure $\omega$ and $\lambda$ is a real number. We also prove that a two-derivation of a connected Poisson manifold $(M,\pi )$ with trivial Poisson cohomology group $H^1(M,\pi )$ is $f\pi$, where $f$ is a Casimir function.

We present here a range of considerations relating to the problem of Dirac for a quantifiable manifold (M, ω) to the dual a Lie algebra with a Poisson structure that is a Lie–Poisson structure minus.