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The usual definition of a twisted Poisson algebra involving a 3-form does not seem to refer to a Poisson algebra that is then twisted, i.e. twisting either the commutative product or the Lie bracket. Have I missed where such a discussion occurs?

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    $\begingroup$ At @RicardoAndrade's request, I have added the tag sg.symplectic-geometry. I have also changed the tag "poissonalgebra" to "poisson-geometry". $\endgroup$
    – Theo Johnson-Freyd
    Commented Nov 13, 2013 at 1:23

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I'm sure experts will be able to provide more detailed answers. My reading of the paper http://arxiv.org/pdf/math/0107133v2.pdf introducing twisted Poisson structures indicates that it is the Courant structure on the generalized tangent bundle which is twisted by the 3-form, and that "twisted Poisson structures" are bivectors defining Dirac structures for the twisted Courant structure.

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  • $\begingroup$ Yes, more precisely the bracket is twisted by adding a term $\endgroup$
    – Jim Stasheff
    Commented Nov 17, 2013 at 21:23
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The (algebraic) Poisson counterpart of Drinfel'd twist deformation from Hopf algebras is the following: let $u$ be an invertible element of a Poisson algebra $A$. Then $A$, with the same associative multiplication but with the new bracket defined by $$ [a, \, b]_u := u^{-1} [ u a, \, u b] $$ is a Poisson algebra called the u-conformal deformation of $A$. I deal with this deformation ( http://arxiv.org/abs/1406.3529 ) and seems to be quite intersting. The geometric meanning of it due by C.M. Marle (1991). A very tempting and difficult question that arise in the context is the classification of all u-conformal deformation of a given Poisson algebra.

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