# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 17th 2015
• (edited Feb 17th 2015)

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)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 23rd 2015
• (edited Feb 23rd 2015)

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

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