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So I'm facing a problem of physical origin which I'd like to understand on a geometric background.

I have a long, tedious bivector involving functional derivatives. I write what it would be the associated bracket over two functions F and G.

It is important to mention that the Schouten-Nijenhius bracket of this bivector $\Lambda$ is not zero, i.e., $[\Lambda,\Lambda]_{SN}\neq 0$, basically, it's not Poisson. And of course because of this the Jacobi identity is not satisfied. I get something of the type:

$\{F,\{G,H\}\}+permutations = \int{f(x,v,t)\nabla B(x)(\partial_vF_{f}\times \partial_vG_{f})\partial_vH_f}d^3xd^3v$

where $f$ is a distribution function, $B$ is a function of only $x$ and $F_f,G_f,H_f$ denote functional derivatives of the functions $F,G,H$ w.r.t. the distribution function $f$.

Any clue of what's the underlying geometric structure. I bet it was a twisted-Poisson structure but it is not. I can't think of any other.

Thanks

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    $\begingroup$ If the Jacobiator (the non-vanishing right-hand side of your formula) does vanish when considered on some cohomology classes (for a suitable notion of cohomology), then it could be the 2-ary bracket of an $L_\infty$ structure. $\endgroup$
    – Igor Khavkine
    Commented Apr 12, 2019 at 8:48
  • $\begingroup$ Thank you Igor. $\endgroup$
    – CristinaSardon
    Commented Apr 15, 2019 at 23:54
  • $\begingroup$ What is missing from the L-infinity picture is how the "triple bracket"(right hand side) comes from a 3-form ( I was looking for a twisted Poisson structure and since they live in Courant algebroids, and there is a formulation by Roytenberg of Courant algebroids in terms of L_infinity algebras). Can you see any connection with a three form? $\endgroup$
    – CristinaSardon
    Commented Apr 15, 2019 at 23:56
  • $\begingroup$ For a (direct) $L_\infty$ interpretation you need the bracket $\{-,-\}$ to be defined on a graded vector space, and the differential on that graded vector space should be a graded derivation of the bracket. In the finite-dimensional Poisson context you can define a Poisson bracket on differential forms such that the exterior derivative is a graded derivation of the bracket (though the grading may be off). Can something like this be adapted? $\endgroup$
    – Ricardo Buring
    Commented Jun 30, 2019 at 15:56

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