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    • 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,ω n+1)(X,\omega_{n+1}), then the Poisson bracket Lie nn-algebra 𝔓𝔬𝔦𝔰(X,ω)\mathfrak{Pois}(X,\omega) from above is an extension of the Lie algebra of Hamiltonian vector fields Vect Ham(X)Vect_{Ham}(X), def. \ref{HamiltonianFormsAndVectorFields} by the cocycle infinity-groupoid H(X,B n1)\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

    H(X,B d1) 𝔓𝔬𝔦𝔰(X,ω) Vect Ham(X,ω) ω[] BH(X,B d1), \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 ω[]\omega[\bullet], when realized on the model of def. \ref{PoissonBracketLienAlgebra}, is degreewise given by by contraction with ω\omega.

    This is FRS13b, theorem 3.3.1.

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

    0H dR d1(X)τ 0𝔓𝔬𝔦𝔰(X,ω)Vect Ham(X)0. 0 \to H^{d-1}_{dR}(X) \longrightarrow \tau_0 \mathfrak{Pois}(X,\omega) \longrightarrow Vect_{Ham}(X) \to 0 \,.
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