Added the statement of the extension theorem:

Given a pre n-plectic manifold $(X,\omega_{n+1})$, then the Poisson bracket Lie $n$-algebra $\mathfrak{Pois}(X,\omega)$ from above is an extension of the Lie algebra of Hamiltonian vector fields $Vect_{Ham}(X)$, def. \ref{HamiltonianFormsAndVectorFields} by the cocycle infinity-groupoid $\mathbf{H}(X,\flat \mathbf{B}^{n-1} \mathbb{R})$ for ordinary cohomology with real number coefficients in that there is a homotopy fiber sequence in the homotopy theory of L-infinity algebras of the form

$\array{ \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R}) &\longrightarrow& \mathfrak{Pois}(X,\omega) \\ && \downarrow \\ && Vect_{Ham}(X,\omega) &\stackrel{\omega[\bullet]}{\longrightarrow}& \mathbf{B} \mathbf{H}(X,\flat \mathbf{B}^{d-1}\mathbb{R}) } \,,$where the cocycle $\omega[\bullet]$, when realized on the model of def. \ref{PoissonBracketLienAlgebra}, is degreewise given by by contraction with $\omega$.

This is FRS13b, theorem 3.3.1.

As a corollary this means that the 0-truncation $\tau_0 \mathfrak{Pois}(X,\omega)$ is a Lie algebra extension by de Rham cohomology, fitting into a short exact sequence of Lie algebras

$0 \to H^{d-1}_{dR}(X) \longrightarrow \tau_0 \mathfrak{Pois}(X,\omega) \longrightarrow Vect_{Ham}(X) \to 0 \,.$ ]]>added to *Poisson bracket Lie n-algebra* the two definitions we have and the statement of their equivalence.

(I am about to edit at *conserved current* and need to point to these ingredients from there)