noncommutative – Page 36 – neverendingbooks

double Poisson algebras

Published October 26, 2004 by lievenlb

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.

2 Comments

anyone interested

Published September 15, 2004 by lievenlb

I've been here before! I mean, I did try to set up
non-commutative algebra&geometry sites before and sooner or later
they always face the same basic problems :

a :
dyspnoea : one person does not have enough fresh ideas
to keep a mathematical site updated daily so that it continues to be of
interest (at least, I'm not one of those who can).

b :
claustrophobia : the topic of non-commutative algebra
& non-commutative geometry is too wide to be covered (cornered) by
one person. More (and differing) views are needed for balance and
continued interest.

c : paranoia : if one is
not entirely naive one has to exercise some restraint trying to protect
ones research plans (or those of students) so the most interesting ideas
never even get posted!

By definition, I cannot solve problems
a) and b) on my own. All I can hope is that, now that the basic
technological problems (such as including LaTeX-code in posts) are
solved, other people are willing to contribute. For this reason I
'depersonalized' this blog : I changed the title, removed all
personal links in the sidebar and so on. I want to open up this site
(but as I said, I've tried this before without much success) to
anyone working in non-commutative algebra and/or non-commutative
geometry who is willing to contribute posts on at least a monthly basis
(or fortnightly, weekly, daily…) for the foreseeable future. At
the moment the following 'categories' of posts are available
(others can be added on request) :

courses : if you want
to tell about your topic of interest in small daily or weekly pieces.
columns : if you want to ventilate an opinion on something
related (even vaguely) to na&g.
nc-algebra : for anything
on non-commutative algebra not in the previous categories.
nc-geometry : for anything on non-commutative geometry not in the
previous categories.
this blog : for suggestions or
explanations on the technology of this site.

Mind you,
I am not looking for people who seek a forum to post
their questions (such people can still add questions as comments to
related posts) but rather for people active in na&g with a personal
opinion on relevance and future of the topic.
If you are
interested in contributing, please email me and we will work
something out. I'll also post information for authors (such as, how
to include tex, how to set restrictions etc.) in a _sticky_ post
soon.

Now, problem c) : in running sites for our master class
on noncommutative geometry I've noticed that some people are more
willing to post lectures notes etc. if they know that there is some
control on who can download their material. For this reason there will
be viewing restrictions on certain posts. Such posts will get a
padlock-sign next to them in the 'recent posts' sidebar (they
will not show up in your main page, if you are not authorized to see
them). I will add another sticky on all of this soon. For now, if you
would only be willing to contribute if there was this safeguard, rest
assured, it will be there soon. All others can of course already sign-up
or wait whether any of these plans (resp. day-dreams) ever work
out….

update (febr 2007) : still waiting
but the padlock idea is abandoned.

cotangent bundles

Published September 9, 2004 by lievenlb

The
previous post in this sequence was [moduli spaces][1]. Why did we spend
time explaining the connection of the quiver
$Q~:~\xymatrix{\vtx{} \ar[rr]^a & & \vtx{} \ar@(ur,dr)^x}$
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_ $\tilde{Q}~:~\xymatrix{\vtx{} \ar@/^/[rr]^a && \vtx{} \ar@(u,ur)^x \ar@(d,dr)_{x^*} \ar@/^/[ll]^{a^*}}$ Clearly,
this already gives away the answer : if the path algebra $C Q$
determines a (non-commutative) manifold $M$ , then the path algebra $C \tilde{Q}$ determines the cotangent bundle of $M$ . Recall that for a
commutative manifold $M$ , the cotangent bundle is the vectorbundle
having at the point $p \in M$ as fiber the linear dual $(T_p M)^*$ of
the tangent space. So, why do we claim that $C \tilde{Q}$
corresponds to the cotangent bundle of $C Q$ ? Fix a dimension vector
$\alpha = (m,n)$ then the representation space
$\mathbf{rep}_{\alpha}~Q = M_{n \times m}(C) \oplus M_n(C)$ is just
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_ $M_{n \times m}(C) \times M_{m \times n}(C) \rightarrow C~:~(A,B) \mapsto tr(AB)$ $M_n(C) \times M_n(C) \rightarrow C~:~(C,D) \mapsto tr(CD)$ and therefore the linear dual
$\mathbf{rep}_{\alpha}~Q^* = M_{m \times n}(C) \oplus M_n(C)$ which is
the representation space $\mathbf{rep}_{\alpha}~Q^s$ of the quiver
$Q^s~:~\xymatrix{\vtx{} & & \vtx{} \ar[ll] \ar@(ur,dr)}$
and therefore we have that the cotangent bundle to the representation
space $\mathbf{rep}_{\alpha}~Q$ $T^* \mathbf{rep}_{\alpha}~Q = \mathbf{rep}_{\alpha}~\tilde{Q}$ Important for us will be that any
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 $C \tilde{Q}$
can be viewed as a non-commutative symplectic manifold with the
symplectic structure determined by the non-commutative 2-form
$\omega = da^* da + dx^* dx$ but before we can define all this we
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 $C Q$ is there also a _tangent bundle_ and does it
again correspond to a new quiver? Well yes, here it is
$\xymatrix{\vtx{} \ar@/^/[rr]^{a+da} \ar@/_/[rr]_{a-da} & & \vtx{} \ar@(u,ur)^{x+dx} \ar@(d,dr)_{x-dx}}$ and the labeling of the
arrows may help you to work through some sections of the Cuntz-Quillen
paper…

[1]: http://www.neverendingbooks.org/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]: http://www.neverendingbooks.org/index.php?p=41
[5]: http://www.arxiv.org/abs/math.QA/0005165
[6]: http://www.arxiv.org/abs/math.AG/0010030

2 Comments

Tag: noncommutative

double Poisson algebras

anyone interested

cotangent bundles