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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 17th 2015
   • (edited Feb 17th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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)

   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)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 23rd 2015
   • (edited Feb 23rd 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded the statement of the [extension theorem](http://ncatlab.org/nlab/show/Poisson+bracket+Lie+n-algebra#ExtensionTheorem): *** Given a [[pre n-plectic manifold]] $(X,\omega_{n+1})$, then the Poisson bracket Lie $n$-algebra $\mathfrak{Pois}(X,\omega)$ from [above](#Definition) is an [[L-infinity algebra cohomology|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](#FRS13b). 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 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

   $$0 \to H^{d-1}_{dR}(X) \longrightarrow \tau_0 \mathfrak{Pois}(X,\omega) \longrightarrow Vect_{Ham}(X) \to 0 \,.$$