Skip to content →

Tag: geometry

TheLibrary (demo)

It is far from finished but you can already visit a demo-version of
TheLibrary which I hope will one day be a useful collection of
online courses and books on non-commutative algebra & geometry. At
the moment it just contains a few of my own things but I do hope that
others will find the format interesting enough to allow me to include
their courses and/or books. So, please try this demo out! But before you
do, make sure that you have a good webbrowser-plugin to view
PDF-documents from within your browser (rather than having to download
the files). If you are using Macintosh 10.3 or better there is a very
nice plugin freely
available whch you only have to drag into your _/Library/Internet
Plug-Ins/_-folder to get it working (after restarting Safari).
If you click on the title you will get a page with hyper-links to all
bookmarks of the pdf-file (for example, if you have used the hyperref package to
(La)TeX your file, you get these bookmarks for free). If you only have a
PDF-file you can always include the required bookmarks using Acrobat.
No doubt the most useful feature (at this moment) of the set-up is
that all files are fully searchable for keywords.
For example, if
you are at the page of my 3 talks on noncommutative
geometry@n
-course and fill out “Azumaya” in the Search
Document-field you will get a screen like the one below

That is, you wlll get all occurrences of 'Azumaya' in
the document together with some of the context as well as page- or
section-links nearby that you can click to get to the paragraph you are
looking for. In the weeks to come I hope to extend the usability of
_TheLibrary_ by offering a one-page view, modular security
enhancements, a commenting feature as well as a popularity count. But,
as always, this may take longer than I want…
If you think
that the present set-up might already be of interest to readers of your
courses or books and if you have a good PDF-file of it available
(including bookmarks) then email and we will try to include your
material!

Leave a Comment

From Galois to NOG


Evariste Galois (1811-1832) must rank pretty high on the all-time
list of moving last words. Galois was mortally wounded in a duel he
fought with Perscheux d\’Herbinville on May 30th 1832, the reason for
the duel not being clear but certainly linked to a girl called
Stephanie, whose name appears several times as a marginal note in
Galois\’ manuscripts (see illustration). When he died in the arms of his
younger brother Alfred he reportedly said “Ne pleure pas, j\’ai besoin
de tout mon courage pour mourir ‚àö‚Ć 20 ans”. In this series I\’ll
start with a pretty concrete problem in Galois theory and explain its
elegant solution by Aidan Schofield and Michel Van den Bergh.
Next, I\’ll rephrase the problem in non-commutative geometry lingo,
generalise it to absurd levels and finally I\’ll introduce a coalgebra
(yes, a co-algebra…) that explains it all. But, it will take some time
to get there. Start with your favourite basefield $k$ of
characteristic zero (take $k = \mathbb{Q}$ if you have no strong
preference of your own). Take three elements $a,b,c$ none of which
squares, then what conditions (if any) must be imposed on $a,b,c$ and $n
\in \mathbb{N}$ to construct a central simple algebra $\Sigma$ of
dimension $n^2$ over the function field of an algebraic $k$-variety such
that the three quadratic fieldextensions $k\sqrt{a}, k\sqrt{b}$ and
$k\sqrt{c}$ embed into $\Sigma$? Aidan and Michel show in \’Division
algebra coproducts of index $n$\’ (Trans. Amer. Math. Soc. 341 (1994),
505-517) that the only condition needed is that $n$ is an even number.
In fact, they work a lot harder to prove that one can even take $\Sigma$
to be a division algebra. They start with the algebra free
product
$A = k\sqrt{a} \ast k\sqrt{b} \ast k\sqrt{c}$ which is a pretty
monstrous algebra. Take three letters $x,y,z$ and consider all
non-commutative words in $x,y$ and $z$ without repetition (that is, no
two consecutive $x,y$ or $z$\’s). These words form a $k$-basis for $A$
and the multiplication is induced by concatenation of words subject to
the simplifying relations $x.x=a,y.y=b$ and $z.z=c$.

Next, they look
at the affine $k$-varieties $\mathbf{rep}(n) A$ of $n$-dimensional
$k$-representations of $A$ and their irreducible components. In the
parlance of $\mathbf{geometry@n}$, these irreducible components correspond
to the minimal primes of the level $n$-approximation algebra $\int(n) A$.
Aidan and Michel worry a bit about reducedness of these components but
nowadays we know that $A$ is an example of a non-commutative manifold (a
la Cuntz-Quillen or Kontsevich-Rosenberg) and hence all representation
varieties $\mathbf{rep}n A$ are smooth varieties (whence reduced) though
they may have several connected components. To determine the number of
irreducible (which in this case, is the same as connected) components
they use _Galois descent
, that is, they consider the algebra $A
\otimes_k \overline{k}$ where $\overline{k}$ is the algebraic closure of
$k$. The algebra $A \otimes_k \overline{k}$ is the group-algebra of the
group free product $\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}
\ast \mathbb{Z}/2\mathbb{Z}$. (to be continued…) A digression : I
cannot resist the temptation to mention the tetrahedral snake problem
in relation to such groups. If one would have started with $4$ quadratic
fieldextensions one would get the free product $G =
\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z} \ast
\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$. Take a supply of
tetrahedra and glue them together along common faces so that any
tertrahedron is glued to maximum two others. In this way one forms a
tetrahedral-snake and the problem asks whether it is possible to make
such a snake having the property that the orientation of the
\’tail-tetrahedron\’ in $\mathbb{R}^3$ is exactly the same as the
orientation of the \’head-tetrahedron\’. This is not possible and the
proof of it uses the fact that there are no non-trivial relations
between the four generators $x,y,z,u$ of $\mathbb{Z}/2\mathbb{Z} \ast
\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z} \ast
\mathbb{Z}/2\mathbb{Z}$ which correspond to reflections wrt. a face of
the tetrahedron (in fact, there are no relations between these
reflections other than each has order two, so the subgroup generated by
these four reflections is the group $G$). More details can be found in
Stan Wagon\’s excellent book The Banach-tarski paradox, p.68-71.

Leave a Comment

neverendingbooks.org

There is a nice, cosy 2nd hand book shop Never Ending
Books
located at 308 Hibiscus Highway, Orewa Beach (New Zealand).
Further, someone registered the domain-name www.neverendingbooks.com and
isn't doing a thing with it at this moment. And that's about it!

As this site will be a strictly non-profit set-up, it made sense
to register the domain-name www.neverendingbooks.org
instead. Partly because many of you seem to find www.matrix.ua.ac.be way too
difficult to remember (judging from the number of times people end up
here Googling _lieven le bruyn_). Unfortunately, registering the
domain-name is the only of three urgent goals I set myself that actually
panned out so far (the other two, _getting a prefix_ and
_partnering up_, won't mean much to you and I'll explain
them later when (if) they work out).
Over the next couple of
weeks it will become gradually clear what this site is all about.
I've worked out things (in theory) over several sleepless nights,
but making them happen will require a lot of extra work.
Oh, you
don't believe I did think some things through? Have a look at the
new header-picture. Recognize those eyes? If you do, you will agree that
this choice was almost forced upon me as I wanted to capture at the same
time the _non-commutative-algebra_, the _non-commutative
geometry_ as well as the _neverending_ aspect of this
site…

Leave a Comment