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+aa^{\star} $ for $a $ an

arrow and $a^{\star} $ 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

$\{ \{ -,- \} \}~:~A \times A

\rightarrow A \otimes A $

which is a derivation in the _second_

argument (for the outer bimodulke structure on $A $) and satisfies

$\{ \{ a,b \} \} = – \{ \{ b,a \} \}^o $ with $~(u \otimes v)^0 = v

\otimes u $

Given such a double bracket one can define an ordinary

bracket (using standard Hopf-algebra notation)

$\{ a,b \} = \sum

\{ \{ 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.

# double Poisson algebras

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