double Poisson algebras

By lieven

non-commutative geometry

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 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 bilinear map
$\{ \{ -,- \} \}~:~A \times A \rightarrow A \otimes A$
which is a derivation in the second argument (for the outer bimodulke structure on ) 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 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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arxiv, geometry, necklace, noncommutative, representations

2 comments
Posted in geometry
Written on Tue, 26 October 2004 at 8:35 am
Tags: arxiv, geometry, necklace, noncommutative, representations
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2 Responses to “double Poisson algebras”

back | neverendingbooks Says:
January 8th, 2008 at 10:32 am
[...] to construct representations of arithmetic groups, a bit like the Granada Notes but with a dash of Double Poisson Algebras to [...]
necklaces (again) | neverendingbooks Says:
January 8th, 2008 at 10:39 am
[...] Schedler's extension of the Lie algebra structure to a Lie bialgebra and its deformation and more recently in connection with Michel Van den Bergh's double Poisson paper. Yesterday, Victor Ginzburg and [...]