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Author: lievenlb

LatexRender plugin for wordpress under Panther

Published August 31, 2004 by lievenlb

After
three days of desperate trial-and-error I seem to have managed to get latexrender working for
wordpress under Mac
OS X.
First things first : if you only want to include some
symbols in your blog-posts the easiest way to do so is to use mimetex and the
corresponding
wordpress-plugin written by Steve Mayer. Follow the
instructions and you will be able to include a limited subset of LaTeX
in your blog within 10 minutes.
If you want more, you have to
work a lot harder. The starting point is to follow Steve’s
blog-entries on latexrender
.
But then under Mac OS X you will probably get error messages
when you activate the plugin. The reason seems to be that most versions
of imagemagick available for
OS X require X-terminal support and PHP gets confused between the two
shells. A typical error message is

Warning:
copy(70afbabac176169545d01f4bd91f3055.gif): failed to open

stream:
No such file or directory in
/Users/lieven/Sites/wordpress/latexrender/class.latexrender.php on
line

269

[Unparseable or potentially dangerous latex
formula. Error 6 ]

As suggested by Steve Mayer there are
two roads to obtain more information on what goes wrong. The first is to
uncomment the _unlink commands _ at the end of the
_class.latexrender.php_ file and look in the _wordpress/latexrender/tmp_
directory for which conversions were done and which failed. The normal
latexrender-procedure is : tex->dvi->ps->gif. Probably you will
get all files but the gifs!

Another (and more useful) source of
informations is to look in the _error-log_ of the Apache-WebServer and
see whether you get things like

This is dvips(k) 5.94a
Copyright 2003 Radical Eye Software (www.radicaleye.com)
\\’
TeX output 2004.08.30:1433\\’ ->
0d48700a5dde6d746813733d26dd8df8.ps

. [1]
sh: line 1:
gs: command not found
convert: no decode delegate for this image
format

/Users/lieven/Sites/weblog/latexrender/tmp/
0d48700a5dde6d746813733d26dd8df8.ps\\’.

convert: missing an image
filename/Users/lieven/Sites/weblog/latexrender/tmp/
0d48700a5dde6d746813733d26dd8df8.gif\\’.

identify: unable to
open image 0d48700a5dde6d746813733d26dd8df8.gif\\': No such file
or directory.
identify: missing an image
filename0d48700a5dde6d746813733d26dd8df8.gif\\’.

`

Here the essential point is that the webserver doesn’t
seem to be able to find GhostScript (even if you have several versions
installed).

To bypass these problems I did two essential
things : (1) in the _class.latexrender.php_ file I rewrote the
conversions so as to use _pdflatex_ instead of tex (to get
immediately a pdf-file rather than the tex->dvi->ps process) and then
use _convert_ to translate this pdf-file into a gif-file. (2) the
version of _convert_ and _include_ (both part of the
ImageMagick package) are those provided by Fink but you should be extremely
careful to install the imagemagick-nox package and not
the imagemagick package! After the command
sudo fink
install imagemagick-nox
you are presented with several
configuration choices. Do _not_ choose on auto-pilot the default
choices but look for options specifying that there is no X-support!
After this, everything should work. If you want to have a look at how
I changed the PHP files, mail
me.

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the one quiver for GL(2,Z)

Published August 26, 2004 by lievenlb

Before the vacation I finished a rewrite of the One quiver to rule them
all note. The main point of that note was to associate to any qurve
$A$ (formerly known as a quasi-free algebra in the terminology of
Cuntz-Quillen or a formally smooth algebra in the terminology of
Kontsevich-Rosenberg) a quiver $Q(A)$ and a dimension vector $\alpha_A$
such that $A$ is etale isomorphic (in a yet to be defined
non-commutative etale toplogy) to a ring Morita equivalent to the path
algebra $lQ(A)$ where the Morita setting is determined by the dimension
vector $\alpha_A$. These “one-quiver settings” are easy to
work out for a group algebra $lG$ if $G$ is the amalgamated free product
of finite groups $G = H_1 \bigstar_H H_2$.

Here is how to do
this : construct a bipartite quiver with the left vertices corresponding
to the irreducible representations of $H_1$, say ${ S_1, .. ,S_k }$ of
dimensions $(d_1, .. ,d_k)$ and the right vertices corresponding to the
irreducible representations of $H_2$, ${ T_1, .. ,T_l }$ of dimensions
$(e_1, .. ,e_l)$. The number of arrows from the $i$-th left vertex to
the $j$-th right vertex is given by the dimension of $Hom_H(S_i,T_j)$
This is the quiver I call the Zariski quiver for $G$ as the finite
dimensional $G$-representations correspond to $\theta$-semistable
representations of this quiver for the stability structure $\theta=(d_1,
.. ,d_k ; -e_1, .. ,-e_l)$. The one-quiver $Q(G)$ has vertices
corresponding to the minimal $\theta$-stable dimension vectors (say
$\alpha,\beta, .. $of the Zariski quiver and with the number of arrows
between two such vertices determined by $\delta_{\alpha
\beta}-\chi(\alpha,\beta)$ where $\chi$ is the Euler form of the Zariski
quiver. In the old note I've included the example of the projective
modular group $PSL_2(Z) = Z_2 \bigstar Z_3$ (which can easily be
generalized to the modular group $SL_2(Z) = Z_4 \bigstar_{Z_2} Z_6$)
which turns out to be the double of the extended Dynkin quiver
$\tilde{A_5}$. In the rewrite I've also included an example of a
congruence subgroup $\Gamma_0(2) = Z_4 \bigstar_{Z_2}^{HNN}$ which is an
HNN-extension. These are somehow the classical examples of interesting
amalgamated (HNN) groups and one would like to have plenty of other
interesting examples. Yesterday I read a paper by Karen Vogtmann called

Automorphisms of free groups and outer space in which I encountered
an amalgamated product decomposition for $GL_2(Z) = D_8 \bigstar_{Z_2
\times Z_2} (S_3 \times Z_2)$where $D_8$ is the diheder group of 8
elements. When I got back from vacation I found a reference to this
result in my mail-box from Warren Dicks. Theorem 23.1, p. 82, in Heiner
Zieschang, Finite Groups of Mapping Classes of Surfaces, LNM 875,
Springer, Berlin, 1981.

I worked out the one-quiver and it has
the somewhat strange form depicted above. It is perfectly possible that
I made mistakes so if you find another result, please let me know.

added material (febr 2007) : mistakes were made and
the correct one quiver can be found elsewhere on this blog.

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the necklace Lie bialgebra

Published June 11, 2004 by lievenlb

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” (see also Victor Ginzburg’s paper “Non-commutative symplectic geometry, quiver
varieties and operads”. 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.

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