# Tag: Ginzburg

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

Now that the preparation for my undergraduate courses in the first semester is more or less finished, I can begin to think about the courses I’ll give this year in the master class
non-commutative geometry. For a change I’d like to introduce the main ideas and concepts by a very concrete example : Ginzburg’s coadjoint-orbit result for the Calogero-Moser space and its
relation to the classification of one-sided ideals in the first Weyl algebra. Not only will this example give me the opportunity to say things about formally smooth algebras, non-commutative
differential forms and even non-commutative symplectic geometry, but it also involves what some people prefer to call _non-commutative algebraic geometry_ (that is the study of graded Noetherian
rings having excellent homological properties) via the projective space associated to the homogenized Weyl algebra. Besides, I have some affinity with this example.

A long time ago I introduced
the moduli spaces for one-sided ideals in the Weyl algebra in Moduli spaces for right ideals of the Weyl algebra and when I was printing a _very_ preliminary version of Ginzburg’s paper
Non-commutative Symplectic Geometry, Quiver varieties, and Operads (probably because he send a preview to Yuri Berest and I was in contact with him at the time about the moduli spaces) the
idea hit me at the printer that the right way to look at the propblem was to consider the quiver

$\xymatrix{\vtx{} \ar@/^/[rr]^a & & \vtx{} \ar@(u,ur)^x \ar@(d,dr)_y \ar@/^/[ll]^b}$

which eventually led to my paper together with Raf Bocklandt Necklace Lie algebras and noncommutative symplectic geometry.

Apart from this papers I would like to explain the following
papers by illustrating them on the above example : Michail Kapranov Noncommutative geometry based on commutator expansions Maxim Kontsevich and Alex Rosenberg Noncommutative smooth
spaces
Yuri Berest and George Wilson Automorphisms and Ideals of the Weyl Algebra Yuri Berest and George Wilson Ideal Classes of the Weyl Algebra and Noncommutative Projective
Geometry
Travis Schedler A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver and of course the seminal paper by Joachim Cuntz and Daniel Quillen on
quasi-free algebras and their non-commutative differential forms which, unfortunately, in not available online.

I plan to write a series of posts here on all this material but I will be very
your name and email is required to do so, you do not have to register and you can even put some latex-code in your post but such a posting will first have to viewed by me to avoid cluttering of
nonsense GIFs in my directories).

Today Travis Schedler posted a nice paper on the arXiv
“A Hopf algebra quantizing a necklace Lie algebra
canonically associated to a quiver”
. I heard the first time about
necklace Lie algebras from Jacques Alev who had heard a talk by Kirillov
who constructed an infinite dimensional Lie algebra on the monomials in
two non-commuting variables X and Y (upto cyclic permutation of the
word, whence ‘necklace’). Later I learned that this Lie algebra was
defined by Maxim Kontsevich for the free algebra in an even number of
variables in his “Formal (non)commutative symplectic geometry” paper
(published in the Gelfand seminar proceedings 1993). Later I extended
this construction together with Raf Bocklandt in “Necklace Lie algebras and non-commutative symplectic
geometry”
. Here, the necklace Lie algebra appears from
(relative) non-commutative differential forms on a symmetric quiver and
its main purpose is to define invariant symplectic flows on quotient
varieties of representations of the quiver.
Travis Schedler
extends this construction in two important ways. First, he shows that
the Lie-algebra is really a Lie-bialgebra hence there is some sort of
group-like object acting on all the representation varieties. Even more
impoprtant, he is able to define a quantization of this structure
defining a Hopf algebra. In this quantization, necklaces play a role
similar to that of (projected) flat links in the plane whereas their
quantization (necklaces with a height) are similar to genuine links in
3-space.
Sadly, at the moment there is no known natural
representations for this Hopf algebra playing a similar role to the
quotient varieties of quiver-varieties in the case of the necklace Lie
bialgebra.