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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,ωn+1), then the Poisson bracket Lie n-algebra 𝔓𝔬𝔦𝔰(X,ω) from above is an extension of the Lie algebra of Hamiltonian vector fields VectHam(X), def. \ref{HamiltonianFormsAndVectorFields} by the cocycle infinity-groupoid H(X,Bn1) 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,Bd1)𝔓𝔬𝔦𝔰(X,ω)VectHam(X,ω)ω[]BH(X,Bd1),

    where the cocycle ω[], when realized on the model of def. \ref{PoissonBracketLienAlgebra}, is degreewise given by by contraction with ω.

    This is FRS13b, theorem 3.3.1.

    As a corollary this means that the 0-truncation τ0𝔓𝔬𝔦𝔰(X,ω) is a Lie algebra extension by de Rham cohomology, fitting into a short exact sequence of Lie algebras

    0Hd1dR(X)τ0𝔓𝔬𝔦𝔰(X,ω)VectHam(X)0.