double Poisson algebras

This morning, Michel Van den Bergh posted an interesting paper on the arXiv entitled Double Poisson Algebras. His main motivation was the construction of a natural Poisson structure on quotient varieties of representations of deformed multiplicative preprojective algebras (introduced by Crawley-Boevey and Shaw in Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem) which he achieves by extending his double Poisson structure on the path algebra of the quiver to the 'obvious' universal localization, that is the one by inverting all for an arrow and its double (the one in the other direction).
For me the more interesting fact of this paper is that his double bracket on the path algebra of a double quiver gives finer information than the necklace Lie algebra as defined in my (old) paper with Raf Bocklandt Necklace Lie algebras and noncommutative symplectic geometry. I will certainly come back to this later when I have more energy but just to wet your appetite let me point out that Michel calls a double bracket on an algebra a bilinear map

which is a derivation in the second argument (for the outer bimodulke structure on ) and satisfies
with
Given such a double bracket one can define an ordinary bracket (using standard Hopf-algebra notation)

which makes into a Loday algebra and induces a Lie algebra structure on . He then goes on to define such a double bracket on the path algebra of a double quiver in such a way that the associated Lie structure above is the necklace Lie algebra.

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