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anyone interested

Published September 15, 2004 by lievenlb

I've been here before! I mean, I did try to set up
non-commutative algebra&geometry sites before and sooner or later
they always face the same basic problems :

a :
dyspnoea : one person does not have enough fresh ideas
to keep a mathematical site updated daily so that it continues to be of
interest (at least, I'm not one of those who can).

b :
claustrophobia : the topic of non-commutative algebra
& non-commutative geometry is too wide to be covered (cornered) by
one person. More (and differing) views are needed for balance and
continued interest.

c : paranoia : if one is
not entirely naive one has to exercise some restraint trying to protect
ones research plans (or those of students) so the most interesting ideas
never even get posted!

By definition, I cannot solve problems
a) and b) on my own. All I can hope is that, now that the basic
technological problems (such as including LaTeX-code in posts) are
solved, other people are willing to contribute. For this reason I
'depersonalized' this blog : I changed the title, removed all
personal links in the sidebar and so on. I want to open up this site
(but as I said, I've tried this before without much success) to
anyone working in non-commutative algebra and/or non-commutative
geometry who is willing to contribute posts on at least a monthly basis
(or fortnightly, weekly, daily…) for the foreseeable future. At
the moment the following 'categories' of posts are available
(others can be added on request) :

  • courses : if you want
    to tell about your topic of interest in small daily or weekly pieces.
  • columns : if you want to ventilate an opinion on something
    related (even vaguely) to na&g.
  • nc-algebra : for anything
    on non-commutative algebra not in the previous categories.
  • nc-geometry : for anything on non-commutative geometry not in the
    previous categories.
  • this blog : for suggestions or
    explanations on the technology of this site.

Mind you,
I am not looking for people who seek a forum to post
their questions (such people can still add questions as comments to
related posts) but rather for people active in na&g with a personal
opinion on relevance and future of the topic.
If you are
interested in contributing, please email me and we will work
something out. I'll also post information for authors (such as, how
to include tex, how to set restrictions etc.) in a _sticky_ post
soon.

Now, problem c) : in running sites for our master class
on noncommutative geometry I've noticed that some people are more
willing to post lectures notes etc. if they know that there is some
control on who can download their material. For this reason there will
be viewing restrictions on certain posts. Such posts will get a
padlock-sign next to them in the 'recent posts' sidebar (they
will not show up in your main page, if you are not authorized to see
them). I will add another sticky on all of this soon. For now, if you
would only be willing to contribute if there was this safeguard, rest
assured, it will be there soon. All others can of course already sign-up
or wait whether any of these plans (resp. day-dreams) ever work
out….

update (febr 2007) : still waiting
but the padlock idea is abandoned.

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differential forms

Published September 14, 2004 by lievenlb

The
previous post in this sequence was [(co)tangent bundles][1]. Let $A$ be
a $V$-algebra where $V = C \times \ldots \times C$ is the subalgebra
generated by a complete set of orthogonal idempotents in $A$ (in case $A
= C Q$ is a path algebra, $V$ will be the subalgebra generated by the
vertex-idempotents, see the post on [path algebras][2] for more
details). With $\overline{A}$ we denote the bimodule quotient
$\overline{A} = A/V$ Then, we can define the _non-commutative
(relative) differential n-forms_ to be $\Omega^n_V~A = A \otimes_V
\overline{A} \otimes_V \ldots \otimes_V \overline{A}$ with $n$ factors
$\overline{A}$. To get the connection with usual differential forms let
us denote the tensor $a_0 \otimes a_1 \otimes \ldots \otimes a_n =
(a_0,a_1,\ldots,a_n) = a_0 da_1 \ldots da_n$ On $\Omega_V~A =
\oplus_n~\Omega^n_V~A$ one defines an algebra structure via the
multiplication $(a_0da_1 \ldots da_n)(a_{n+1}da_{n+2} \ldots da_k)$$=
\sum_{i=1}^n (-1)^{n-i} a_0da_1 \ldots d(a_ia_{i+1}) \ldots da_k$
$\Omega_V~A$ is a _differential graded algebra_ with differential $d :
\Omega^n_V~A \rightarrow \Omega^{n+1}_V~A$ defined by $d(a_0 da_1 \ldots
da_n) = da_0 da_1 \ldots da_n$ This may seem fairly abstract but in
case $A = C Q$ is a path algebra, then the bimodule $\Omega^n_V~A$ has a
$V$-generating set consisting of precisely the elements $p_0 dp_1
\ldots dp_n$ with all $p_i$ non-zero paths in $A$ and such that
$p_0p_1 \ldots p_n$ is also a non-zero path. One can put another
algebra multiplication on $\Omega_V~A$ which Cuntz and Quillen call the
_Fedosov product_ defined for an $n$-form $\omega$ and a form $\mu$ by
$\omega Circ \mu = \omega \mu -(-1)^n d\omega d\mu$ There is an
important relation between the two structures, the degree of a
differential form puts a filtration on $\Omega_V~A$ (with Fedosov
product) such that the _associated graded algebra_ is $\Omega_V~A$ with
the usual product. One can visualize the Fedosov product easily in the
case of path algebras because $\Omega_V~C Q$ with the Fedosov product is
again the path algebra of the quiver obtained by doubling up all the
arrows of $Q$. In our basic example when $Q$ is the quiver
$\xymatrix{\vtx{} \ar[rr]^u & & \vtx{} \ar@(ur,dr)^v} $ the
algebra of non-commutative differential forms equipped with the Fedosov
product is isomorphic to the path algebra of the quiver
$\xymatrix{\vtx{} \ar@/^/[rr]^{a=u+du} \ar@/_/[rr]_{b=u-du} & &
\vtx{} \ar@(u,ur)^{x=v+dv} \ar@(d,dr)_{y=v-dv}} $ with the
indicated identification of arrows with elements from $\Omega_V~C Q$.
Note however that we usually embed the algebra $C Q$ as the degree zero
differential forms in $\Omega_V~C Q$ with the usual multiplication and
that this embedding is no longer an algebra map (but a based linear map)
for the Fedosov product. For this reason, Cuntz and Quillen invent a
Yang-Mills type argument to “flow” this linear map to an algebra
embedding, but to motivate this we will have to say some things about
[curvatures][3].

[1]: http://www.neverendingbooks.org/index.php?p=352
[2]: http://www.neverendingbooks.org/index.php?p=349
[3]: http://www.neverendingbooks.org/index.php?p=353

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some smaller steps

Published September 13, 2004 by lievenlb

It
always amazes me how much time I have to waste in trying to get
tech-stuff (such as this weblog) working the way I want. You will barely
notice it but again I spend too much time delving in PHP-scripts,
sometimes with minor success, most of the time almost wrecking this
weblog…

An example : it took me a day to figure out why
this page said there was just 1 visitor online whereas log files showed
otherwise. The PHP-script I used checked this by looking at the
IP-address via _REMOTE_ADDR_ which is perfectly OK on an ordinary
Mac OS 10.3 machine, but _not_ on an OS X-Server! For some reason
it gives as the REMOTE_ADDR just the IP address of the Server (that
is, www.matrix.ua.ac.be in this case) so whoever came by this page got
tagged as 143.129.75.209 and so the script thought there was just one
person around… The trivial way around it is changing every
occurence of REMOTE_ADDR by _HTTP_PC_REMOTE_ADDR_.
Easy enough but it took me a while to figure it out.

Another
example : over the week-end this weblog got a stalker! There were over
100 hits from 38.113.198.9, so whoever that is really liked this site
but didn't have time to read a thing… Again, the standard
solution is to ban the IP-address and most weblog-packages have such a
tool on their admin-page. But whathever I tried and Googled WordPress doesn't seem to have it
on board. There were a few hacks and plugins around claiming to do
something about it but none of them worked! So, I tried more drastic
actions such as editing .htaccess files which I thought would solve
everything (again, no problem under 10.3 but _not_ under
10.3-Server!). Once more, a couple of hours lost trying to figure out
how to get the firewall of a Mac-Server do what I needed. The upshot is
that I know now all dark secrets of the _ipfw_ command, so no
more stalking around this site…

In the process of
grounding my stalker, I decided that I needed better site-stats than my
homemade log-file provided. Fortunetely, this time I picked a package
that worked without too much hassle (one more time I had to make the
REMOTE_ADDR substitution but apart from that all went well). You will
see not too much of the power of this stats-package on the page (apart
from the global counter), I feel that such things are best forgotten
until something strange occurs (like stalkers, spammers and other
weirdos). A nice side-effect though was that for the first time I had a
look at _referring pages_, that is the URL leading to this weblog.
Lots of Google searches (some strange ones) but today there were also a
number of referrals from a Chinese blog. I checked it out and it turned
out to be the brand new Math is Math! Life is Life! weblog…

Another time
consuming thing was getting the BBC-news RSS feeds working in the
sidebar, so that you still get _some_ feel for reality while
being trapped here. I am not yet satisfied with the layout under
Explorer, but then everyone should move on to Safari (so I did give up
trying to work out the PHP-script).

But most time I wasted on
something that so far has left no trace whatsoever here. A plugin that
allows specific posts to be read only by registered users (of a certain
'level', that is WordPress can give users a level from 0 to 10
with specific degrees of freedom). But clearly at the same time I wanted
the rest of the world to have at least some indication of what they were
missing (such as a title with a nice padlock next to it) but so far I
didn't get it working. The only trace of a closed posting would be
in the sidebar-listing of the ten last posts but gives an error message
when an unauthorized user clicks on it. So, still a lot of
headache-sensitive work left to do, but it is about time to get back to
mathematics…

update (febr. 2007) : the
padlock-idea is abandoned.

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