The
arXiv is a bit like cable tv : on certain days there seems to be nothing
interesting on, whereas on others it’s hard to decide what to see in
real time and what to record for later. Today was one of the better
days, at least on the arXiv. Pavel Etingof submitted the
notes of a course he gave at ETH in the spring and summer of 2005 Lectures on
Calogero-Moser systems. I always sympathize with people taking time
to explain what they are interested in to non-experts, especially if
they even take more time to write up course notes so that the rest of us
can also benefit from these talks. Besides, it is always more rewarding
to learn a topic from a key-figure such as Etingof, rather than sitting
through talks on this given by people who only embrace a topic as a
career move. However, as I’m no longer that much into Calogero-Moser
stuff I’ve put Pavel’s notes in recording mode as I definitely have to
spend some time getting through that other paper posted today : Notes on A-infinity
algebras, A-infinity categories and non-commutative geometry. I by
Maxim
Kontsevich and Yan
Soibelman. They really come close to things that interest me right
now and although I’m not the greatest coalgebra-fan, they may give me
just enough reasons to bite the bullet. On a different topic : with
plenty of help from Jacques Distler, my
neverending planet
is now also serving MathML, but you need to view it using Firefox and
have all the required fonts
installed.
Tag: non-commutative
Let us
take a hopeless problem, motivate why something like non-commutative
algebraic geometry might help to solve it, and verify whether this
promise is kept.
Suppose we want to know all solutions in invertible
matrices to the braid relation (or Yang-Baxter equation)
All such solutions (for varying size of matrices)
form an additive Abelian category 
, so a big step forward would be to know all its
simple solutions (that is, those whose matrices cannot be brought in
upper triangular block form). A literature check shows that even this
task is far too ambitious. The best result to date is the classification
due to Imre Tuba and
Hans Wenzl of simple solutions of which the matrix size is at most
5.
For fixed matrix size n, finding solutions in is the same as solving a system of 
cubic
polynomial relations in 
unknowns, which quickly becomes a daunting task. Algebraic geometry
tells us that all solutions, say 
form an affine closed subvariety of -dimensional affine space. If we assume that is a smooth variety (that is, a manifold) and
if we know one solution explicitly, then we can use the tangent space in
this point to linearize the problem and to get at all solutions in a
neighborhood.
So, here is an idea : assume that itself would be a non-commutative manifold, then
we might linearize our problem by considering tangent spaces and obtain
new solutions out of already known ones. But, what is a non-commutative
manifold? Well, by the above we at least require that for all integers n
the commutative variety is a commutative manifold.
But, there
is still some redundancy in our problem : if 
is a
solution, then so is any conjugated pair 
where 
is a basechange matrix. In categorical terms, we are only
interested in isomorphism classes of solutions. Again, if we fix the
size n of matrix-solutions, we consider the affine variety as a variety with a 
-action
and we like to classify the orbits of simple solutions. If is a manifold then the theory of Luna slices
provides a method, both to linearize the problem as well as to reduce
its complexity. Instead of the tangent space we consider the normal
space N to the -orbit
(in a suitable solution). On this affine space, the stabilizer subgroup
acts and there is a natural one-to-one
correspondence between -orbits
in and -orbits in the normal space N (at least in a
neighborhood of the solution).
So, here is a refinement of the
idea : we would like to view as a non-commutative manifold with a group action
given by the notion of isomorphism. Then, in order to get new isoclasses
of solutions from a constructed one we want to reduce the size of our
problem by considering a linearization (the normal space to the orbit)
and on it an easier isomorphism problem.
However, we immediately
encounter a problem : calculating ranks of Jacobians we discover that
already 
is not a smooth variety so there is not a
chance in the world that might be a useful non-commutative manifold.
Still, if is a
solution to the braid relation, then the matrix 
commutes with both X and Y.
If is a
simple solution, this means that after performing a basechange, 
becomes a scalar matrix, say 
. But then, 
is a solution to
and all such solutions form a
non-commutative closed subvariety, say 
of and if we know all (isomorphism classes of)
simple solutions in we have solved our problem as we just have to
bring in the additional scalar 
.
Here we strike gold : is indeed a non-commutative manifold. This can
be seen by identifying 
with one of the most famous discrete infinite groups in mathematics :
the modular group 
. The modular group acts by Mobius
transformations on the upper half plane and this action can be used to
write as the free group product 
. Finally, using
classical representation theory of finite groups it follows that indeed
all 
are commutative manifolds (possibly having
many connected components)! So, let us try to linearize this problem by
looking at its non-commutative tangent space, if we can figure out what
this might be.
Here is another idea (or rather a dogma) : in the
world of non-commutative manifolds, the role of affine spaces is played
by 
the representations of finite quivers Q. A quiver
is just on oriented graph and a representation of it assigns to each
vertex a finite dimensional vector space and to each arrow a linear map
between the vertex-vector spaces. The notion of isomorphism in is of course induced by base change actions in all
of these vertex-vector spaces. (to be continued)
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…
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The third volume in the NeverEndingBooks-series will be written by Geert Van de Weyer and will
be about (double) Poisson structures in the noncommutative world. Volume
4&5 are becoming clearer every day and if you think you have a
project fitting in this series, you can always email to
[info@neverenedingbooks.org][3].
As for the NeverEndingBooks-URL, I will
probably close this blog by the end of the month (at its first
birthday). The main reason is that I found out that it takes several
people to maintain a mathematical blog for some time. So, if you want to
co-author a group-blog on noncommutative algebra and/or noncommutative
geometry, please [email me][5] (or even better, leave a comment here so
that other people may be willing to join in too) and if there is enough
critical mass to go ahead with the plan I will be happy to set up a
group-blog at noncommutative.org.
At
this URL I’ll probably put a frontpage for the book-series we started
and which you can buy at all times via lulu.com/neverendingbooks. It will contain errata- and suggestions-pages for each volume and
details about forthcoming books, links etc… Btw. it would probably
be a good commercial move to delete TheLibrary links sooner, now that
even String Theorists are driven to this site via
Lazariou’s paper On
the non-commutative geometry of topological D-branes
As my
main objective next year will be to write courses (from first year down
to post-doc level) I will set up (again) a Moodle site (mainly in English,
although UA-students will be free to add to it in Dutch). News about it
will be posted eventually at my regular, but forgotten
homepage and perhaps here.
Once again, if you are interested
to contribute at unregular intervals to a noncommutative group-blog,
please leave a comment!
[3]: mailto: info@neverendingbooks.org
[5]: mailto: lieven.lebruyn@ua.ac.be
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](http://www.neverendingbooks.org/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 $X$ and spice it up to get a very non-commutative
qurve. First, we bring in finite groups. Let $G$ be a finite group
acting on $X$, then we can form the skew-group algebra $A = \mathbfk[X]
\bigstar G$. These are examples of prime Noetherian qurves (aka
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 $A =
(\mathbf k[X] \bigstar G) \ast_{\mathbf k G} \mathbfk H$ where $G
\subset H$ is an interesting extension of finite groups. Then, $A$ is
again a qurve on which $H$ acts in a way compatible with the $G$-action
on $X$ and $A$ is hugely non-commutative… A very basic example :
let $\mathbb{Z}/2\mathbb{Z}$ act on the affine line $\mathbfk[x]$ by
sending $x \mapsto -x$ and consider a finite [simple
group](http://mathworld.wolfram.com/SimpleGroup.html) $M$. As every
simple group has an involution, we have an embedding
$\mathbb{Z}/2\mathbb{Z} \subset M$ and can construct the qurve
$A=(\mathbfk[x] \bigstar \mathbb{Z}/2\mathbb{Z}) \ast_{\mathbfk
\mathbb{Z}/2\mathbb{Z}} \mathbfk M$ on which the simple group $M$ acts
compatible with the involution on the affine line. To study the
corresponding non-commutative manifold, that is the Abelian category
$\mathbf{rep}~A$ of all finite dimensional representations of $A$ we have
to compute the [one quiver to rule them
all](http://www.matrix.ua.ac.be/master/coursenotes/onequiver.pdf) for
$A$. Because $A$ is a qurve, all its representation varieties
$\mathbf{rep}_n~A$ are smooth affine varieties, but they may have several
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 $Ext^1_A(S_i,S_j)$ where $S_i,S_j$ are simple
$A$-representations lying in the respective components. All this
may seem hard to compute but it can be reduced to the study of another
quiver, the Zariski quiver associated to $A$ which is a bipartite quiver
with on the left the \’one quiver\’ for $\mathbfk[x] \bigstar
\mathbb{Z}/2\mathbb{Z}$ which is just $\xymatrix{\vtx{}
\ar@/^/[rr] & & \vtx{} \ar@/^/[ll]} $ (where the two vertices
correspond to the two simples of $\mathbb{Z}/2\mathbb{Z}$) and on the
right the \’one quiver\’ for $\mathbf k M$ (which just consists of as
many verticers as there are simple representations for $M$) and where
the number of arrows from a left- to a right-vertex is the number of
$\mathbb{Z}/2\mathbb{Z}$-morphisms between the respective simples. To
make matters even more concrete, let us consider the easiest example
when $M = A_5$ the alternating group on $5$ letters. The corresponding
Zariski quiver then turns out to be $\xymatrix{& & \vtx{1} \\\
\vtx{}\ar[urr] \ar@{=>}[rr] \ar@3[drr] \ar[ddrr] \ar[dddrr] \ar@/^/[dd]
& & \vtx{4} \\\ & & \vtx{5} \\\ \vtx{} \ar@{=>}[uurr] \ar@{=>}[urr]
\ar@{=>}[rr] \ar@{=>}[drr] \ar@/^/[uu] & & \vtx{3} \\\ & &
\vtx{3}} $ The Euler-form of this quiver can then be used to
calculate the dimensions of the EXt-spaces giving the number of arrows
in the \’one quiver\’ for $A$. To find the vertices, that is, the
generators of the component semigroup we have to find the minimal
integral solutions to the pair of equations saying that the number of
simple $\mathbb{Z}/2\mathbb{Z}$ components based on the left-vertices is
equal to that one the right-vertices. In this case it is easy to see
that there are as many generators as simple $M$ representations. For
$A_5$ they correspond to the dimension vectors (for the Zariski quiver
having the first two components on the left) $\begin{cases}
(1,2,0,0,0,0,1) \\ (1,2,0,0,0,1,0) \\ (3,2,0,0,1,0,0) \\
(2,2,0,1,0,0,0) \\ (1,0,1,0,0,0,0) \end{cases}$ We now have all
info to determine the \’one quiver\’ for $A$ and one would expect a nice
result. Instead one obtains a complete graph on all vertices with plenty
of arrows. More precisely one obtains as the one quiver for $A_5$
$\xymatrix{& & \vtx{} \ar@{=}[dll] \ar@{=}[dddl] \ar@{=}[dddr]
\ar@{=}[drr] & & \\\ \vtx{} \ar@(ul,dl)|{4} \ar@{=}[rrrr]|{6}
\ar@{=}[ddrrr]|{8} \ar@{=}[ddr]|{4} & & & & \vtx{} \ar@(ur,dr)|{8}
\ar@{=}[ddlll]|{6} \ar@{=}[ddl]|{10} \\\ & & & & & \\\ & \vtx{}
\ar@(dr,dl)|{4} \ar@{=}[rr]|{8} & & \vtx{} \ar@(dr,dl)|{11} & } $
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 $\mathbf{iss}_n~A$ have a local Poisson
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 $PSL_2(\mathbb{F}_7)$ acts) when the situation was sexed-up
to the sporadic simple Mathieu group $M_{24}$ (of which
$PSL_2(\mathbb{F}_7)$ is a maximal subgroup).
If you recognize where this picture was
taken, you will know that I\’m back from France. If you look closer you
will see two bikes, my own Bulls mountainbike
in front and Stijn\’s
lightweight bike behind.
If you see the relative position of the
saddles, you will know that Stijn is at least 20 cm taller. Let me add
that he is also at least 20 yrs. younger and 20 kgs. stronger and it
will be clear that I had a hard (but fun) time trying to follow him
uphill. Btw. this picture (and the next dozen or so) was taken by Jan and I\’ll try to add the next
days a couple of shots he likes more.
Since then I\’ve been
writing up a paper which I hope will be ready to put online by
september. It\’s all about using non-commutative geometry to construct
representations of arithmetic groups, a bit like the Granada Notes but with a dash of
Double Poisson
Algebras to it.
A positive outcome of this short break is
a renewed interest in the NeverEndingBooks project, but more on this
later. For now, let me just add that Raf
decided to feed my noncommutative geometry@n (version 2)
to a printing on demand publisher. So, if you want a perfect bound
paperback version of it (for 12 Euro approx.) you\’d better email him at once (at the
moment he will order just 5 copies).
I’m always
extremely slow to pick up a trend (let alone a hype), in mathematics as
well as in real life. It took me over a year to know of the existence of
_blogs_ and to realize that they were a much easier way to
maintain a webpage than manually modifying HTML-pages. But, eventually I
sometimes get there, usually with the help of the mac-dev-center. So, once again,
I read their gettings things done with your mac article long after it was
posted and completely unaware of the Getting Things Done (or GTD) hype.
At first, it just
sounds as one of those boring managament-nonsense-peptalk things (and
probably that is precisely what it generically is). Or what do you think
about the following resume from Getting
started with ‘Getting things done’ :
- identify all the
stuff in your life that isnÕt in the right place (close all open
loops) - get rid of the stuff that isnÕt yours or you donÕt
need right now - create a right place that you trust and that
supports your working style and values - put your stuff in the
right place, consistently - do your stuff in a way that honors
your time, your energy, and the context of any given moment - iterate and refactor mercilessly
But in fact there is
also some interesting material around at the 43 folders website which bring this
management-talk closer to home such as the How does a
nerd hack GTD? post.
Also of interest are his findings after
a year working with the GTD setup. These are contained in three posts :
A Year
of Getting Things Done: Part 1, The Good Stuff, followed by A Year of
Getting Things Done: Part 2, The Stuff I Wish I Were Better At to
end with A Year of
Getting Things Done: Part 3, The Future of GTD?. If these three
postings don’t get you intrigued, nothing else will.
So, is
there something like _GMD : Getting Mathematics Done_? Clearly, I
don’t mean getting theorems proved, that’s a thing of a few seconds of
inspiration and months to fill in the gaps. But, perhaps all this GTD
and the software mentioned can be of some help to manage the
everyday-workflow of mathematicians, such as checking the arXiv and the
web, maintaining an email-, pdf- and BiBTeX-database, drafting papers,
books and courses etc.
In the next few weeks I’ll try out some
of the tricks. Probably another way to state this is the question “which
Apps will survive Tiger?” Now that it is official that Tiger (that is, Mac
10.4 to non-apple eaters) will be released by the end of the month it is
time to rethink which of the tools I really like to keep and which is
just useless garbage I picked up along the road. For example, around
this time last year I had a Perl
phase and bought half a meter or so of O’Reilly Perl-books. And yes
I did write a few simple scripts, some useful such as my own arXiv RSS-feeds,
some not so useful as a web-spider I wrote to check on changes in the
list of hamepages of people working in non-commutative algebra and
geometry. A year later I realize I’ll never become a Perl Monk. So from now on I want to
make my computer-life as useful and easy as possible, relying on wizards
to provide me with cool software to use and help me enjoy mathematics
even more. I’ll keep you posted how my GMD-adventure goes.
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;