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Tag: quivers

TheLibrary

Published April 25, 2005 by lievenlb

Some
people objected to the set-up of TheLibrary because it was serving only one-page at a
time. They’d rather have a longer download-time if they can then
browse through the paper/book, download it and print if they decide to
do so.

Fine! Today I spend some hours refilling TheLibrary with
texts. As before you are able to search any document for specific words
(as explained in elsewhere )
and click on any section or page to view the wanted material in your
browser (assuming you have the proper plugin installed). This time I
used pdfscreen to make the notes more readable on your
screen.

If you prefer, you can download the text, safe it on
your hard-disk and browse at leasure. Whereas these versions are
intended to be read from screen, you can also print them if you have to
at 150dpi. In the next couple of weeks I hope to add some material :
older and undergraduate courses and I’ll add a _papers_ section
where I’ll put all my papers of which I can recover a TeX-file. For
starters, I included the revision of the Qurves and quivers
paper in the courses-folder.

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pdfsync

Published April 13, 2005 by lievenlb

I
expect to be writing a lot in the coming months. To start, after having
given the course once I noticed that I included a lot of new material
during the talks (mainly concerning the component coalgebra and some
extras on non-commutative differential forms and symplectic forms) so
I\’d better update the Granada notes
soon as they will also be the basis of the master course I\’ll start
next week. Besides, I have to revise the Qurves and
Quivers-paper and to start drafting the new bachelor courses for
next academic year (a course on representation theory of finite groups,
another on Riemann surfaces and an upgrade of the geometry-101 course).

So, I\’d better try to optimize my LaTeX-workflow and learn
something about the pdfsync package.
Here is what it is supposed to do :

pdfsync is
an acronym for synchronization between a pdf file and the TeX or so
source file used in the production process. As TeX system is not a
WYSIWYG editor, you cannot modify the output directly, instead, you must
edit a source file then run the production process. The pdfsync helps
you finding what part of the output corresponds to what line of the
source file, and conversely what line of the source file corresponds to
a location of a given page in the ouput. This feature is achieved with
the help of an auxiliary file: foo.pdfsync corresponding to a foo.pdf.

All you have to do is to put the pdfsync.sty file
in the directory _~/Library/texmf/tex/latex/pdfsync.sty_ and to
include the pdfsync-package in the preamble of the LaTeX-document. Under
my default iTex-front-end TeXShop it
works well to go from a spot in the PDF-file to the corresponding place
in the source-code, but in the other direction it only shows the
appropriate page rather than indicate the precise place with a red dot
as it does in the alternative front-end iTeXMac.

A major
drawback for me is that pdfsync doesn\’t live in harmony with my
favorite package for drawing commutative diagrams diagrams.sty. For example, the 75 pages of the current
version of the Granada notes become blown-up to 96 pages because each
commutative diagram explodes to nearly page size! So I will also have to
translate everything to xymatrix&#
8230;

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necklaces (again)

Published March 23, 2005 by lievenlb

I have
been posting before on the necklace Lie algebra : on Travis
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 Travis Schedler posted their paper Moyal quantization of
necklace Lie algebras on the arXiv in which they give a Moyal-type
construction of the Hopf algebra deformation of the necklace Lie
bialgebra found by Schedler last year.
It would be nice if
someone worked out a few examples of these constructions in full detail.
But as often in the case of (wild) quiver situation it is not clear what
an 'interesting' example might be. For the finite and tame case
we have a full classification by (extended) Dynkin diagrams so a natural
class of examples but it isn't clear how to find gems in the
complement.
One natural source of double quiver situations seems
to come from what I called the One Quiver of a
formally smooth algebra. This one quiver of group algebras of some
interesting arithemetical groups such as the modular group
$PSL_2(\mathbb{Z}) $ and $SL_2(\mathbb{Z}) $ were calculated before and
turned out to be consisting of one (resp. two) components which are the
double of the tame quiver $\tilde{A}_5 $.
To obtain the double of
a wild quiver situation loook at the group $GL_2(\mathbb{Z}) = D_4
\bigstar_{D_2} D_6 $. In a previous post
I thought to have calculated it, but lately I found that this was
incorrect. Even the version I computed last week still had some mistakes
as Raf
Bocklandt discovered. But as of yesterday we are pretty certain that
the one quiver for $GL_2(\mathbb{Z}) $ consists of two components. One of
these is the double quiver of an interesting wild quiver

$\xymatrix{& \vtx{} \ar@{=}[rr] \ar@{=}[dd] & & \vtx{} \ar@{=}[dd]
\\ \vtx{} \ar@{=}[ur] \ar@{=}[rr] \ar@{=}[dd] & & \vtx{} \ar@{.}[ur]
\ar@{.}[dd] \ar@{=}[dr] \\ & \vtx{} \ar@{.}[rr] \ar@{=}[dr] & & \vtx{}
\\ \vtx{} \ar@{=}[rr] \ar@{.}[ur] & & \vtx{} \ar@{=}[ur]} $

where each double line indicates that there is an arrow in each
direction between the vertices. So, it is an interwoven pattern of one
big cycle of length 6 (reminiscent of the modular group case) with 4
cycles of length 5. Perhaps the associated necklace Lie (bi)algebra and
its deformation might be interesting to work out.
However, the
second component of the one quiver for $GL_2(\mathbb{Z}) $ is _not_
symmetric.Maybe I will come back to the calculation of these quivers
later.

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