cotangent bundles

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_ Clearly,
this already gives away the answer : if the path algebra $CQ$
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 $(T_{p}M{)}^{\ast }$ of
the tangent space. So, why do we claim that $C\tilde{Q}$
corresponds to the cotangent bundle of $CQ$? 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_ $M_{n\times m}(C)\times M_{m\times n}(C)\to C:(A,B)\mapsto tr(AB)$ $M_n(C)\times M_n(C)\to C:(C,D)\mapsto tr(CD)$ and therefore the linear dual
${\mathbf{rep}}_{\alpha }{Q}^{\ast }=M_{m\times n}(C)\oplus M_n(C)$ which is
the representation space ${\mathbf{rep}}_{\alpha }{Q}^s$ of the quiver

and therefore we have that the cotangent bundle to the representation
space ${\mathbf{rep}}_{\alpha }Q$ $T^{\ast }{\mathbf{rep}}_{\alpha }Q={\mathbf{rep}}_{\alpha }\tilde{Q}$ 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
$\omega =d{a}^{\ast }da+d{x}^{\ast }dx$ 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 $CQ$ is there also a _tangent bundle_ and does it
again correspond to a new quiver? Well yes, here it is
and the labeling of the
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