Multi-derivations of Poisson algebras

Abstract

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 [Formula: see text] 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 [Formula: see text] and multi-derivations of the linear Poisson manifold [Formula: see text], and show that a two-derivation of a connected symplectic manifold [Formula: see text] is [Formula: see text], where [Formula: see text] is the Poisson bivector field induced by the symplectic structure [Formula: see text] and [Formula: see text] is a real number. We also prove that a two-derivation of a connected Poisson manifold [Formula: see text] with trivial Poisson cohomology group [Formula: see text] is [Formula: see text], where [Formula: see text] is a Casimir function.