Klein’s
quartic
objects around. For example, it is a Hurwitz curve meaning that the
finite group of symmetries (when the genus is at least two this group
can have at most
the case of the quartic is
simple group of that order,
Klein\’s group. John Baez has written a [beautiful page](http://math.ucr.edu/home/baez/klein.html) on the Klein quartic and
its symmetries. Another useful source of information is a paper by Noam
Elkies [The Klein quartic in number theory](www.msri.org/publications/books/Book35/files/elkies.pd).
The quotient map
branch points of orders
coordinates
non-free
Now, remove from
subset
form the Klein stack (or hereditary order)
the skew group algebra. In case the open subset
non-free orbits, the [one quiver](lievenlb.local/master/coursenotes/onequiver.pdf) of this
qurve has the following shape
non-free orbits and the vertices correspond to the isoclasses of simple
two such of dimension
dimension
\’trinity\’ and \’the dwarfs\’. As we want to spice up later this
Klein stack to a larger group, we need to know the structure of these
exceptional simples as
written a paper on the general problem of finding the
simples of skew-group algebras
reference please let me know. I used an old paper by Idun Reiten and
Christine Riedtmann to do this case (which is easier as the stabilizer
subgroups are cyclic and hence the induced representations of their
one-dimensionals correspond to the exceptional simples).
neverendingbooks Posts
I got
Tracks working under Tiger by trial and error starting with a suggestion
from Jan who found a fix here. From this I stumbled on for an hour or so on my iMac
untill near the very end I found that the tracks-page itself now
has a Tiger-section. So, let’s try to do it all over again from fresh on
my regular machine (a little iBook).
Start with
this page and read it all the way to the comments. There is a
comment by Jamie on installing Rails and MySQL on a fresh Tiger which
looks like the thing I want followed immediately by another post by Jose
Marinez on
Installing Ruby on Rails on Tiger which I decided to follow by the
letter (with one noticeable exception!). So first I downloaded the Rails
installer on Tiger. Next, an install of the Standard version of
MySQL. I used version 4.0 for OS
X 10.3 !!!! NOT 4.1 !!!!. I installed MySQL and the StartUpItem.
Next, I opened Terminal and typed the following commands
– cd
/usr/local/mysql
– sudo chown -R mysql data/
– sudo echo
– sudo ./bin/mysqld_safe &
and verified it by performing a
simple test
/usr/local/mysql/bin/mysql test
Next, I
secured everything by having a root-password
/usr/local/mysql/bin/mysqladmin -u root password
__*
Then I remembered that I’d better
not have to type the whole path so I did a
echo ‘export
PATH=/usr/local/mysql/bin:$PATH’ >> ~/.bash_profile
Also, I want
to admin MySQL via phpmyadmin so I
installed it. Then I enabled my webserver to use PHP using this post though I
did use vi rather than pico! Next, i did a check whether everything
worked fine by typing in safari
http://localhost/~lieven/phpmyadmin/index.php
Followed by
a
sudo gem install mysql –
–with-mysql-dir=/usr/local/mysql
Now, it is time to get the Tracks-package
and to follow these instructions And, it works! Here is the proof :
Here the
story of an idea to construct new examples of non-commutative compact
manifolds, the computational difficulties one runs into and, when they
are solved, the white noise one gets. But, perhaps, someone else can
spot a gem among all gibberish…
[Qurves](https://lievenlb.local/toolkit/pdffile.php?pdf=/TheLibrary/papers/qaq.pdf) (aka quasi-free algebras, aka formally smooth
algebras) are the \’affine\’ pieces of non-commutative manifolds. Basic
examples of qurves are : semi-simple algebras (e.g. group algebras of
finite groups), [path algebras of
quivers](http://www.lns.cornell.edu/spr/2001-06/msg0033251.html) and
coordinate rings of affine smooth curves. So, let us start with an
affine smooth curve
qurve. First, we bring in finite groups. Let
acting on
hereditary orders). A more pompous way to phrase this is that these are
precisely the [one-dimensional smooth Deligne-Mumford
stacks](http://www.math.lsa.umich.edu/~danielch/paper/stacks.pdf).
As the 21-st century will turn out to be the time we discovered the
importance of non-Noetherian algebras, let us make a jump into the
wilderness and consider the amalgamated free algebra product
again a qurve on which
on
let
sending
group](http://mathworld.wolfram.com/SimpleGroup.html)
simple group has an involution, we have an embedding
compatible with the involution on the affine line. To study the
corresponding non-commutative manifold, that is the Abelian category
to compute the [one quiver to rule them
all](https://lievenlb.local/master/coursenotes/onequiver.pdf) for
connected components. The direct sum of representations turns the set of
all these components into an Abelian semigroup and the vertices of the
\’one quiver\’ correspond to the generators of this semigroup whereas
the number of arrows between two such generators is given by the
dimension of
may seem hard to compute but it can be reduced to the study of another
quiver, the Zariski quiver associated to
with on the left the \’one quiver\’ for
correspond to the two simples of
right the \’one quiver\’ for
many verticers as there are simple representations for
the number of arrows from a left- to a right-vertex is the number of
make matters even more concrete, let us consider the easiest example
when
Zariski quiver then turns out to be
calculate the dimensions of the EXt-spaces giving the number of arrows
in the \’one quiver\’ for
generators of the component semigroup we have to find the minimal
integral solutions to the pair of equations saying that the number of
simple
equal to that one the right-vertices. In this case it is easy to see
that there are as many generators as simple
having the first two components on the left)
info to determine the \’one quiver\’ for
result. Instead one obtains a complete graph on all vertices with plenty
of arrows. More precisely one obtains as the one quiver for
with the number of arrows (in each direction) indicated. Not very
illuminating, I find. Still, as the one quiver is symmetric it follows
that all quotient varieties
structure. Clearly, the above method can be generalized easily and all
examples I did compute so far have this \’nearly complete graph\’
feature. One might hope that if one would start with very special
curves and groups, one might obtain something more interesting. Another
time I\’ll tell what I got starting from Klein\’s quartic (on which the
simple group
to the sporadic simple Mathieu group