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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 1+aaโ‹† for a an
arrow and aโ‹† 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 a bilinear map
{{โˆ’,โˆ’}} : Aร—Aโ†’AโŠ—A
which is a derivation in the _second_
argument (for the outer bimodulke structure on A) and satisfies
{{a,b}}=โ€“{{b,a}}o with  (uโŠ—v)0=vโŠ—u
Given such a double bracket one can define an ordinary
bracket (using standard Hopf-algebra notation)
{a,b}=โˆ‘{{a,b}}(1){{a,b}}(2)
which makes A into
a Loday
algebra
and induces a Lie algebra structure on A/[A,A]. 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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