Tag: geometry

get your brain subscribed to

Published February 8, 2006 by lieven

In the
‘subscribe
to my brain’ post I promised to blog on how-to get your own

button up and running on your homepage. It seems rather unlikely
that I’ll ever keep that promise if I don’t do it right away. So, here
we go for a quick tour :

step 1 : set up a rudimentary
FoaF-file : read the FoaF post if
you dont know what it’s all about. The easiest way to get a simple
FoaF-file of your own is to go to the FoaF-a-matic
webpage and fill in the details you feel like broadcasting over the
web, crucial is your name and email information (for later use) but
clearly the more details you fill out and the more Friends you add the
more useful your file becomes. Click on the ‘foaf-me’ button and
copy the content created. Observe that there is no sign of my email
adress, it is encrypted in the _mbox_sha1sum_ data. Give this
file a name like _foaf.rdf_ or _myname.rdf_ and put it on
your webserver to make it accessible. Also copy your
_mbox_sha1sum_ info for later smushing.

step 2 : subscribe to online services and modify your
online-life accordingly : probably you have already a few of
these accounts, but if not, take a free subscription just for fun and
(hopefully) later usage to the following sites :

del.icio.us a social bookmarks manager
citeUlike a service to
organise your academic papers
connotea a reference management
service for scientists
bloglines a web-based personal news
aggregator
43things a
‘What do you want to do with your life?’ service
audioscrobbler a database that
tracks listening habits and does wonderful things with statistics
backpackit a ‘be better organized’ service (Update october 2017 : Tom Howard emails: “I thought I’d reach out because we’ve just updated our guide which reviews the best alternatives to Backpack. Here’s the link”
flickr an online photo management and
sharing application
technorati a Google-for-weblogs
upcoming a social event
calendar
webjay a playlist
community

So far, I’m addicted to del.icio.us and use
citeUlike but hardly any of the others (but I may come back to this
later). The great thing about these services is that you get more
value-information back if you feed more into the system. For example, if
you use del.icio.us as your ‘public’ bookmarks-file you get to
know how many other people have bookmarked the same site and you can
access their full bookmarks which often is a far more sensible way to
get at the information you are after than mindless Googling. So, whereas
I was at first a bit opposed to the exhibisionist-character of these
services (after all, anyone with web-access can have a look at
‘your’ info), I’ve learned that the ‘social’ feature of
these services can be beneficial to get the right information I want.
Hence, the hardest part is not to get an account with these services but
to adopt your surfing behavior in such a way that you maximize this
added value. And, as I mentioned before, I’m doing badly myself but hope
that things will improve…

step 3 : turn these
accounts into an OPML file : Knowing the URL of your foaf-file
and sha1-info (step 1) and your online accounts, go to the FOAF Online Account
Description Generator and feed it with your data. You will then get
another foaf-file back (save the source in a file such as
_accounts.rdf_ and put it on your webserver). Read the Lost Boy’s
posts Subscribe to my
brain and foaf:
OnlineAccount Generator for more background info. Then, use the SubscribeToMyBrain-
form to get an OPML-file out of the account.rdf file and your sha1.
Save the source as _mybrain.opml_.

step 4 :
add/delete information you want : The above method uses generic
schemes to deduce relevant RSS-data from an account name, which works
for some services, but doesn’t for all. So, if you happen to know the
URL of RSS-feeds for one of these services, you can always add it
manually to the OPML-file (or delete data you don’t want to
publish…). My own attitude is to make all public web-data
available and to leave it to the subscriber to unsubscribe those parts
of my brain (s)he is not interested in. I know there are people whoo are
mainly interested to find out whether I put another paper online, would
tolerate some weblog-posts but have no interest in my musical tast,
whereas there are others who would like me to post more on 43things,
flickr or upcoming and don’t give a damn about my mathematics…
Apart from these online subscriptions, it is also a good idea to include
additional RSS-feeds you produce, such as those of your weblog or use my
Perl
script to have your own arXiv-feeds.

step 5 : make
your ‘subscribe to my brain’-button : Now, put the
OPML-file on your webserver, put the button

on your
homepage and link it to the file. Also, add information on your site,
similar to the one I gave in my own
subscription post so that your readers know what to do when do want
to subscribe to (parts of) your brain. Finally, (and optionally though
I’d wellcome it) send me an email with your URL so that I can subscribe
(next time you’re in Antwerp I’ll buy you a beer) and for the first few
who do so and are working in noncommutative geometry and/or
noncommutative algebra, I’ll send a copy of a neverending book. Mind
you, this doesn’t apply to local people, I’m already subscribed to their
brain on a daily basis…

a noncommutative topology 2

Published February 1, 2006 by lieven

A *qurve*
is an affine algebra such that $~\Omega^1~A$ is a projective
$~A~$-bimodule. Alternatively, it is an affine algebra allowing lifts of
algebra morphisms through nilpotent ideals and as such it is the ‘right’
noncommutative generalization of Grothendieck’s smoothness criterium.
Examples of qurves include : semi-simple algebras, coordinate rings of
affine smooth curves, hereditary orders over curves, group algebras of
virtually free groups, path algebras of quivers etc.

Hence, qurves
behave a lot like curves and as such one might hope to obtain one day a
‘birational’ classification of them, if we only knew what we mean
by this. Whereas the etale classification of them is understood (see for
example One quiver to
rule them all or Qurves and quivers )
we don’t know what the Zariski topology of a qurve might be.

Usually, one assigns to a qurve $~A~$ the Abelian category of all its
finite dimensional representations $\mathbf{rep}~A$ and we would like to
equip this set with a topology of sorts. Because $~A~$ is a qurve, its
scheme of n-dimensional representations $\mathbf{rep}_n~A$ is a smooth
affine variety for each n, so clearly $\mathbf{rep}~A$ being the disjoint
union of these acquires a trivial but nice commutative topology.
However, we would like open sets to hit several of the components
$\mathbf{rep}_n~A$ thereby ‘connecting’ them to form a noncommutative
topological space associated to $~A~$.

In a noncommutative topology on
rep A I proposed a way to do this and though the main idea remains a
good one, I’ll ammend the construction next time. Whereas we don’t know
of a topology on the whole of $\mathbf{rep}~A$, there is an obvious
ordinary topology on the subset $\mathbf{simp}~A$ of all simple finite
dimensional representations, namely the induced topology of the Zariski
topology on $~\mathbf{spec}~A$, the prime spectrum of twosided prime ideals
of $~A~$. As in commutative algebraic geometry the closed subsets of the
prime spectrum consist of all prime ideals containing a given twosided
ideal. A typical open subset of the induced topology on $\mathbf{simp}~A$
hits many of the components $\mathbf{rep}_n~A$, but how can we extend it to
a topology on the whole of the category $\mathbf{rep}~A$ ?

Every
finite dimensional representation has (usually several) Jordan-Holder
filtrations with simple successive quotients, so a natural idea is to
use these filtrations to extend the topology on the simples to all
representations by restricting the top (or bottom) of the Jordan-Holder
sequence. Let W be the set of all words w such as $U_1U_2 \ldots U_k$
where each $U_i$ is an open subset of $\mathbf{simp}~A$. We can now define
the *left basic open set* $\mathcal{O}_w^l$ consisting of all finite
dimensional representations M having a Jordan-Holder sequence such that
the i-th simple factor (counted from the bottom) belongs to $U_i$.
(Similarly, we can define a *right basic open set* by counting from the
top or a *symmetric basic open set* by merely requiring that the simples
appear in order in the sequence). One final technical (but important)
detail is that we should really consider equivalence classes of left
basic opens. If w and w’ are two words we will denote by $\mathbf{rep}(w
\cup w’)$ the set of all finite dimensional representations having a
Jordan-Holder filtration with enough simple factors to have one for each
letter in w and w’. We then define $\mathcal{O}^l_w \equiv
\mathcal{O}^l_{w’}$ iff $\mathcal{O}^l_w \cap \mathbf{rep}(w \cup w’) =
\mathcal{O}^l_{w’} \cap \mathbf{rep}(w \cup w’)$. Equivalence classes of
these left basic opens form a partially ordered set (induced by
set-theoretic inclusion) with a unique minimal element 0 (the empty set
corresponding to the empty word) and a uunique maximal element 1 (the
set $\mathbf{rep}~A$ corresponding to the letter $w=\mathbf{simp}~A$).
Set-theoretic union induces an operation $\vee$ and the operation
$~\wedge$ is induced by concatenation of words, that is,
$\mathcal{O}^l_w \wedge \mathcal{O}^l_{w’} \equiv \mathcal{O}^l_{ww’}$.
This then defines a **left noncommutative topology** on $\mathbf{rep}~A$ in
the sense of Van Oystaeyen (see [part
1](https://lievenlb.local/index.php/noncommutative-topology-1 $
). To be precise, it satisfies the axioms in the left and middle column
of the following picture and
similarly, the right basic opens give a right noncommutative topology
(satisfying the axioms of the middle and right columns) whereas the
symmetric opens satisfy all axioms giving the basis of a noncommutative
topology. Even for very simple finite dimensional qurves such as
$\begin{bmatrix} \mathbb{C} & \mathbb{C} \\ 0 & \mathbb{C}
\end{bmatrix}$ this defines a properly noncommutative topology on the
Abelian category of all finite dimensional representations which
obviously respect isomorphisms so is really a noncommutative topology on
the orbits. Still, while this may give a satisfactory local definition,
in gluing qurves together one would like to relax simple representations
to *Schurian* representations. This can be done but one has to replace
the topology coming from the Zariski topology on the prime spectrum by
the partial ordering on the *bricks* of the qurve, but that will have to
wait until next time…

noncommutative topology (1)

Published January 29, 2006 by lieven

A couple of days ago Ars Mathematica had a post Cuntz on noncommutative topology pointing to a (new, for me) paper by Joachim Cuntz

A couple of years ago, the Notices of the AMS featured an article on noncommutative geometry a la Connes: Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz. The hallmark of this approach is the heavy reliance on K theory. The first few pages of the article are fairly elementary (and full of intriguing pictures), before the K theory takes over.

A few comments are in order. To begin, the paper is **not** really about noncommutative geometry a la Connes, but rather about noncommutative geometry a la Cuntz&Quillen (based on quasi-free algebras) or, equivalently, a la Kontsevich (formally smooth algebras) or if I may be so bold a la moi (qurves).

About the **intruiging pictures** : it seems to be a recent trend in noncommutative geometry research papers to include meaningless pictures to lure the attention of the reader. But, unlike aberrations such as the recent pastiche by Alain Connes and Mathilde Marcolli A Walk in the Noncommutative Garden, Cuntz is honest about their true meaning

I am indebted to my sons, Nicolas and Michael,
for the illustrations to the examples above. Since
these pictures have no technical meaning, they
are only meant to provide a kind of suggestive
visualization of the corresponding quantum spaces.

As one of these pictures made it to the cover of the **Notices** an explanation was included by the cover-editor

About the Cover :

The image on this month’s cover arose from
Joachim Cuntz’s effort to render into visible art
his own internal vision of a noncommutative
torus, an object otherwise quite abstract. His
original idea was then implemented by his son
Michael in a program written in Pascal. More
explicitly, he says that the construction started
out with a triangle in a square, then translated
the triangle by integers times a unit along a line
with irrational slope; plotted the images thus
obtained in a periodic manner; and stopped
just before the figure started to seem cluttered.
Many mathematicians carry around inside
their heads mental images of the abstractions
they work with, and manipulate these objects
somehow in conformity with their mental imagery. They probably also make aesthetic judgements of the value of their work according to
the visual qualities of the images. These presumably common phenomena remain a rarely
explored domain in either art or psychology.

—Bill Casselman(covers@ams.org)

There can be no technical meaning to the pictures as in the Connes and Cuntz&Quillen approach there is only a noncommutative algebra and _not_ an underlying geometric space, so there is no topology, let alone a noncommutative topology. Of course, I do understand why Cuntz&others name it as such. They view the noncommutative algebra as the ring of functions on some virtual noncommutative space and they compute topological invariants (such as K-groups) of the algebras and interprete them as information about the noncommutative topology of these virtual and unspecified spaces.

Still, it is perfectly possible to associate to a qurve (aka quasi-free algebra or formally smooth algebra) a genuine noncommutative topological space. In this series of posts I’ll explain the little I know of the history of this topic, the thing I posted about it a couple of years ago, why I abandoned the project and the changes I made to it since and the applications I have in mind, both to new problems (such as the birational_classification of qurves) as well as classical problems (such as rationality problems for $PGL_n $ quotient spaces).

Although others have tried to define noncommutative topologies before, I learned about them from Fred Van Oystaeyen. Fred spend the better part of his career constructing structure sheaves associated to noncommutative algebras, mainly to prime Noetherian algebras (the algebras of preference for the majority of non-commutative algebraists). So, suppose you have an ordinary (meaning, the usual commutative definition) topological space X associated to this algebra R, he wants to define an algebra of sections on every open subset $X(\sigma) $ by taking a suitable localization of the algebra $Q_{\sigma}(R) $. This localization is taken with respect to a suitable filter of left ideals $\mathcal{L}(\sigma) $ of R and is defined to be the subalgebra of the classiocal quotient ring $Q(R) $ (which exists because $R$ is prime Noetherian in which case it is a simple Artinian algebra)

$Q_{\sigma}(R) = { q \in Q(R)~|~\exists L \in \mathcal{L}(\sigma)~:~L q \subset R } $

(so these localizations are generalizations of the usual Ore-type rings of fractions). But now we come to an essential point : if we want to glue this rings of sections together on an intersection $X(\sigma) \cap X(\tau) $ we want to do this by ‘localizing further’. However, there are two ways to do this, either considering $~Q_{\sigma}(Q_{\tau}(R)) $ or considering $Q_{\tau}(Q_{\sigma}(R)) $ and these two algebras are only the same if we impose fairly heavy restrictions on the filters (or on the algebra) such as being compatible.

As this gluing property is essential to get a sheaf of noncommutative algebras we seem to get stuck in the general (non compatible) case. Fred’s way out was to make a distinction between the intersection $X_{\sigma} \cap X_{\tau} $ (on which he put the former ring as its ring of sections) and the intersection $X_{\tau} \cap X_{\sigma} $ (on which he puts the latter one). So, the crucial new ingredient in a noncommutative topology is that the order of intersections of opens matter !!!

Of course, this is just the germ of an idea. He then went on to properly define what a noncommutative topology (and even more generally a noncommutative Grothendieck topology) should be by using this localization-example as guidance. I will not state the precise definition here (as I will have to change it slightly later on) but early version of it can be found in the Antwerp Ph.D. thesis by Luc Willaert (1995) and in Fred’s book Algebraic geometry for associative algebras.

Although _qurves_ are decidedly non-Noetherian (apart from trivial cases), one can use Fred’s idea to associate a noncommutative topological space to a qurve as I will explain next time. The quick and impatient may already sneak at my old note a non-commutative topology on rep A but please bear in mind that I changed my mind since on several issues…