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
arrow and
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
which is a derivation in the _second_
argument (for the outer bimodulke structure on
Given such a double bracket one can define an ordinary
bracket (using standard Hopf-algebra notation)
which makes
a Loday
algebra and induces a Lie algebra structure on
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.
Tag: noncommutative
The
previous post in this sequence was [moduli spaces][1]. Why did we spend
time explaining the connection of the quiver
to moduli spaces of vectorbundles on curves and moduli spaces of linear
control systems? At the start I said we would concentrate on its _double
quiver_
this already gives away the answer : if the path algebra
determines a (non-commutative) manifold
commutative manifold
having at the point
the tangent space. So, why do we claim that
corresponds to the cotangent bundle of
an affine space so in its point the tangent space is the representation
space itself. To define its linear dual use the non-degeneracy of the
_trace pairings_
the representation space
and therefore we have that the cotangent bundle to the representation
space
cotangent bundle has a natural _symplectic structure_. For a good
introduction to this see the [course notes][2] “Symplectic geometry and
quivers” by [Geert Van de Weyer][3]. As a consequence
can be viewed as a non-commutative symplectic manifold with the
symplectic structure determined by the non-commutative 2-form
will have to recall some facts on non-commutative differential forms.
Maybe [next time][4]. For the impatient : have a look at the paper by
Victor Ginzburg [Non-commutative Symplectic Geometry, Quiver varieties,
and Operads][5] or my paper with Raf Bocklandt [Necklace Lie algebras
and noncommutative symplectic geometry][6]. Now that we have a
cotangent bundle of
again correspond to a new quiver? Well yes, here it is
arrows may help you to work through some sections of the Cuntz-Quillen
paper…
[1]: https://lievenlb.local/index.php?p=39
[2]: http://www.win.ua.ac.be/~gvdwey/lectures/symplectic_moment.pdf
[3]: http://www.win.ua.ac.be/~gvdwey/
[4]: https://lievenlb.local/index.php?p=41
[5]: http://www.arxiv.org/abs/math.QA/0005165
[6]: http://www.arxiv.org/abs/math.AG/0010030