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Tag: non-commutative

why nag? (3)

Published March 29, 2005 by lievenlb

Here is
the construction of this normal space or chart \mathbf{chart}_{\Gamma}. The sub-semigroup of Z^5 (all
dimension vectors of Q) consisting of those vectors \alpha=(a_1,a_2,b_1,b_2,b_3) satisfying the numerical condition a_1+a_2=n=b_1+b_2+b_3 is generated by six dimension vectors,
namely those of the 6 non-isomorphic one-dimensional solutions in \mathbf{rep}~\Gamma

S_1 = \xymatrix@=.4cm{ & & & & \vtx{1} \\ \vtx{1} \ar[rrrru]^1 \ar[rrrrd] \ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{0} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}} \qquad S_2 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\& & & & \vtx{1} \\\vtx{1} \ar[rrrruuu] \ar[rrrru]^1 \ar[rrrrd] & & & & \\ & & & & \vtx{0}}

S_3 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{1} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd]^1 & & & & \\ & & & & \vtx{0} \\ \vtx{0} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{1}} \qquad S_4 = \xymatrix@=.4cm{ & & & & \vtx{1} \\ \vtx{0} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1} \ar[rrrruuu]^1 \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}}

S_5 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{1} \ar[rrrru] \ar[rrrrd]^1 \ar[rrrrddd] & & & & \\ & & & & \vtx{1} \\ \vtx{0} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd] & & & & \\ & & & & \vtx{0}} \qquad S_6 = \xymatrix@=.4cm{ & & & & \vtx{0} \\ \vtx{0} \ar[rrrru] \ar[rrrrd] \ar[rrrrddd] & & & & \\ & & & & \vtx{0} \\ \vtx{1} \ar[rrrruuu] \ar[rrrru] \ar[rrrrd]^1 & & & & \\ & & & & \vtx{1}}

In
particular, in any component \mathbf{rep}_{\alpha}~Q containing an open subset of
representations corresponding to solutions in \mathbf{rep}~\Gamma we have a particular semi-simple solution

M = S_1^{\oplus g_1} \oplus S_2^{\oplus g_2} \oplus S_3^{\oplus g_3} \oplus S_4^{\oplus g_4} \oplus S_5^{\oplus g_5} \oplus S_6^{\oplus g_6}

and in
particular \alpha = (g_1+g_3+g_5,g_2+g_4+g_6,g_1+g_4,g_2+g_5,g_3+g_6). The normal space
to the GL(\alpha)-orbit of M in \mathbf{rep}_{\alpha}~Q can be identified with the representation
space \mathbf{rep}_{\beta}~Q where \beta=(g_1,\ldots,g_6) and Q is the quiver of the following
form

\xymatrix{ & \vtx{g_1} \ar@/^/[ld]^{C_{16}} \ar@/^/[rd]^{C_{12}} & \\ \vtx{g_6} \ar@/^/[ru]^{C_{61}} \ar@/^/[d]^{C_{65}} & & \vtx{g_2} \ar@/^/[lu]^{C_{21}} \ar@/^/[d]^{C_{23}} \\ \vtx{g_5} \ar@/^/[u]^{C_{56}} \ar@/^/[rd]^{C_{54}} & & \vtx{g_3} \ar@/^/[u]^{C_{32}} \ar@/^/[ld]^{C_{34}} \\ & \vtx{g_4} \ar@/^/[lu]^{C_{45}} \ar@/^/[ru]^{C_{43}} & }

and we can
even identify how the small matrices C_{ij} fit
into the 3 \times 2 block-decomposition of the base-change matrix B

B = \begin{bmatrix} \begin{array}{ccc|ccc} 1_{a_1} & 0 & 0 & C_{21} & 0 & C_{61} \\ 0 & C_{34} & C_{54} & 0 & 1_{a_4} & 0 \\ \hline C_{12} & C_{32} & 0 & 1_{a_2} & 0 & 0 \\ 0 & 0 & 1_{a_5} & 0 & C_{45} & C_{65} \\ \hline 0 & 1_{a_3} & 0 & C_{23} & C_{43} & 0 \\ C_{16} & 0 & C_{56} & 0 & 0 & 1_{a_6} \\ \end{array} \end{bmatrix}

Hence, it makes sense
to call Q the non-commutative normal space to the isomorphism problem in
\mathbf{rep}~\Gamma. Moreover, under this correspondence simple
representations of Q (for which both the dimension vectors and
distinguishing characters are known explicitly) correspond to simple
solutions in \mathbf{rep}~\Gamma.

Having completed our promised
approach via non-commutative geometry to the classification problem of
solutions to the braid relation, it is time to collect what we have
learned. Let \beta=(g_1,\ldots,g_6) with n = \gamma_1 + \ldots + \gamma_6, then for every
non-zero scalar \lambda \in \mathbb{C}^* the matrices

X = \lambda B^{-1} \begin{bmatrix} 1_{g_1+g_4} & 0 & 0 \\ 0 & \rho^2 1_{g_2+g_5} & 0 \\ 0 & 0 & \rho 1_{g_3+g_6} \end{bmatrix} B \begin{bmatrix} 1_{g_1+g_3+g_5} & 0 \\ 0 & -1_{g_2+g_4+g_6} \end{bmatrix}

Y = \lambda \begin{bmatrix} 1_{g_1+g_3+g_5} & 0 \\ 0 & -1_{g_2+g_4+g_6} \end{bmatrix} B^{-1} \begin{bmatrix} 1_{g_1+g_4} & 0 & 0 \\ 0 & \rho^2 1_{g_2+g_5} & 0 \\ 0 & 0 & \rho 1_{g_3+g_6} \end{bmatrix} B

give a solution of size
n to the braid relation. Moreover, such a solution can be simple only if
the following numerical relations are satisfied

g_i \leq g_{i-1} + g_{i+1}

where indices are viewed
modulo 6. In fact, if these conditions are satisfied then a sufficiently
general representation of Q does determine a simple solution in \mathbf{rep}~B_3 and conversely, any sufficiently general simple n
size solution of the braid relation can be conjugated to one of the
above form. Here, by sufficiently general we mean a Zariski open (hence
dense) subset.

That is, for all integers n we have constructed
nearly all (meaning a dense subset) simple solutions to the braid
relation. As to the classification problem, if we have representants of
simple \beta-dimensional representations of the quiver Q, then the corresponding
solutions (X,Y) of
the braid relation represent different orbits (up to finite overlap
coming from the fact that our linearizations only give an analytic
isomorphism, or in algebraic terms, an etale map). Such representants
can be constructed for low dimensional \beta.
Finally, our approach also indicates why the classification of
braid-relation solutions of size \leq 5 is
easier : from size 6 on there are new classes of simple
Q-representations given by going round the whole six-cycle!

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why nag? (2)

Published March 28, 2005 by lievenlb

Now, can
we assign such an non-commutative tangent space, that is a \mathbf{rep}~Q for some quiver Q, to \mathbf{rep}~\Gamma? As \Gamma = \mathbb{Z}_2 \ast \mathbb{Z}_3 we may
restrict any solution V=(X,Y)
in \mathbf{rep}~\Gamma to the finite subgroups \mathbb{Z}_2 and \mathbb{Z}_3. Now, representations of finite cyclic groups are
decomposed into eigen-spaces. For example

V \downarrow_{\mathbb{Z}_2} = V_+ \oplus V_-

where V_{\pm} = \{ v \in V~|~g.v = \pm v \} with g the
generator of \mathbb{Z}_2. Similarly,

V \downarrow_{\mathbb{Z}_3} = V_1 \oplus V_{\rho} \oplus V_{\rho^2}

where \rho is a
primitive 3-rd root of unity. That is, to any solution V \in \mathbf{rep}~\Gamma we have found 5 vector spaces V_+,V_-,V_1,V_{\rho} and V_{\rho^2} so we would like them to correspond to the vertices
of our conjectured quiver Q.

What are the arrows of Q, or
equivalently, is there a natural linear map between the vertex-vector
spaces? Clearly, as

V_+ \oplus V_- = V = V_1 \oplus V_{\rho} \oplus V_{\rho^2}

any choice of two bases of V (one
compatible with the left-side decomposition, the other with the
right-side decomposition) are related by a basechange matrix B which we
can decompose into six blocks (corresponding to the two decompositions
in 2 resp. 3 subspaces

B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \\ B_{31} & B_{32} \end{bmatrix}

which gives us 6 linear maps between the
vertex-vector spaces. Hence, to V \in \mathbf{rep}~\Gamma does correspond in a natural way a
representation of dimension vector \alpha=(a_1,a_2,b_1,b_2,b_3) (where dim(V_+)=a_1,\ldots,dim(V_{\rho^2})=b_3) of the quiver Q which
is of the form

\xymatrix{ & & & & \vtx{b_1} \\ \vtx{a_1} \ar[rrrru]^(.3){B_{11}} \ar[rrrrd]^(.3){B_{21}} \ar[rrrrddd]_(.2){B_{31}} & & & & \\ & & & & \vtx{b_2} \\ \vtx{a_2} \ar[rrrruuu]_(.7){B_{12}} \ar[rrrru]_(.7){B_{22}} \ar[rrrrd]_(.7){B_{23}} & & & & \\ & & & & \vtx{b_3}}

Clearly, not every representation of \mathbf{rep}~Q is obtained in this way. For starters, the
eigen-space decompositions force the numerical restriction

a_1+a_2 = dim(V) = b_1+b_2+b_3

on the
dimension vector and the square matrix constructed from the arrow-linear
maps must be invertible. However, if both these conditions are
satisfied, we can reconstruct the (isomorphism class) of the solution in
\mathbf{rep}~\Gamma from this quiver representation by taking

X = B^{-1} \begin{bmatrix} 1_{b_1} & 0 & 0 \\ 0 & \rho^2 1_{b_2} & 0 \\ 0 & 0 & \rho 1_{b_3} \end{bmatrix} B \begin{bmatrix} 1_{a_1} & 0 \\ 0 & -1_{a_2} \end{bmatrix}

Y = \begin{bmatrix} 1_{a_1} & 0 \\ 0 & -1_{a_2} \end{bmatrix} B^{-1} \begin{bmatrix} 1_{b_1} & 0 & 0 \\ 0 & \rho^2 1_{b_2} & 0 \\ 0 & 0 & \rho 1_{b_3} \end{bmatrix} B

Hence, it makes sense to
view \mathbf{rep}~Q as a linearization of, or as a tangent space to,
\mathbf{rep}~\Gamma. However, though we reduced the study of
solutions of the polynomial system of equations to linear algebra, we
have not reduced the isomorphism problem in size. In fact, if we start
of with a matrix-solution V=(X,Y)
of size n we end up with a quiver-representation of total dimension 2n.
So, can we construct some sort of non-commutative normal space to the
isomorphism classes? That is, is there another quiver Q whose
representations can be interpreted as normal-spaces to orbits in certain
points?

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seen this quiver?

Published March 25, 2005 by lievenlb

The above quiver on 10 vertices is not symmetric, but has the
interesting property that every vertex has three incoming and three
outgoing arrows. If you have ever seen this quiver in another context,
please drop me a line. My own interest for it is that it is the ‘one
quiver’ for a non-commutative compactification of $GL_2(\mathbb{Z}) $. If
you like to know what I mean by this, you might consult the
Granada-notes which I hope to post over the weekend.

On a
different matter, if you want to know what all this hype on derived
categories and the classification project is about but got lost in the
pile of preprints, you might have a look at the Bourbaki talk by Raphael
Rouquier Categories
derivees et geometrie birationnelle posted today on the arXiv.

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B for bricks

Published January 28, 2005 by lievenlb

Last time we
argued that a noncommutative variety might be an _aggregate_
which locally is of the form $\mathbf{rep}~A$ for some affine (possibly
non-commutative) $C$-algebra $A$. However, we didn't specify what we
meant by 'locally' as we didn't define a topology on
$\mathbf{rep}~A$, let alone on an arbitrary aggregate. Today we will start
the construction of a truly _non-commutative topology_ on
$\mathbf{rep}~A$.
Here is the basic idea : we start with a thick
subset of finite dimensional representations on which we have a natural
(ordinary) topology and then we extend this to a non-commutativce
topology on the whole of $\mathbf{rep}~A$ using extensions. The impatient
can have a look at my old note A noncommutative
topology on rep A but note that we will modify the construction here
in two essential ways.
In that note we took $\mathbf{simp}~A$, the
set of all fnite dimensional simple representations, as thick subset
equipped with the induced Zariski topology on the prime spectrum
$\mathbf{spec}~A$. However, this topology doesn't behave well with
respect to the gluings we have in mind so we will extend $\mathbf{simp}~A$
substantially.

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writing

Published January 23, 2005 by lievenlb

A long
time ago Don Passman
told me the simple “secret” for writing books : “Get up and,
before you do anything else, try to write 2 or 3 pages. If you do this
every day, by the end of the year you’ll have a pretty thick book.”

Probably the best advice ever for those who need to get a thesis or book
finished. I’ve managed to live by this rule for several months in a
row (the first half of 2000 leading to version 2 and the winter of 2001-2002
resulting in version 3) and I can recommend it to
anyone in need for some (self)dicipline. It feels just like training,
hard in the beginning but after a couple of weeks you’re addicted.
Also the pitfalls are similar. On certain days you have so much energy
that it is easy to write 10 or more pages (or in the revision process,
to revise 30 or more pages). Don’t do it! Tomorrow you will be
exhausted and you will not be able to do a single page but you will
convince yourself that it is not needed as you did more than enough the
day before. And you’ll feel and say the same thing the day after, and
the next day! and before you realize it you’ll be way behind
schedule. So, rule 1 : do 2 pages mimimun, 3 or 4 if possible but never
more than 5!

Another useful bit of advice comes from
Lewis Caroll’s ‘Through the looking glass’ in which the Red
King says

Start at the beginning, then continue until
you reach the end. Then stop.

Too many bookprojects
never get past the planning stages. It is much more fun to dream up the
perfect book than it is to write the first paragraph. Also, when the
writing on chapter X goes slow, it is tempting to begin with chapter X+1
or any other chapter that seems like more fun, and before you know
you’ll end up with a complete mess (and believe me, I know what I’m
talking about here).

Armed with these two guiding rules I began
the new year writing version pi of my book. (Oh, a marginal note : some
people seem to think that I set up ‘NeverEndingBooks’ to get my
book published. It may surely be the case that I’ll get _a_
book published there, but _the_ book I promised already a long
time ago to the EMS-publishing
house! So, if you have an interesting bookproject for
‘NeverEndingBooks’ please contact us.) Anyway, the writing goes
slow! I’m already far behind schedule. So far I produced just over 20
pages! Part of the problem is that I want the book to be self-contained
and from past experiences with our ‘masterclass non-commutative
geometry’ I know that this means including a lot of elementary
material (it seems that sudents are eager on entering a masterclass on
non-commutative geometry without knowing the basics of either
non-commutative algebra or algebraic geometry). So. I started out with
believe it or not the definition of matrix-multiplication! But the book
has a pretty steap learning curve, by page 3 I’m already using
Grassmannians to classify left ideals in matrix-algebras! But I was
surprised how long it took me to come up with my own proofs of all this
‘trivial’ material. But the main problem is : lack of motivation.
I’m no longer convinced that one has to write technical books to aid
the younger generation. They are already far too technical!Perhaps it
would be far better to write books helping to develop creativity? But
how? And why are there so few of such books around. In fact, I know of
only one book trying to achieve this : An Invitation to General
Algebra and Universal Constructions By George Bergman. His chapter 0
‘about the course and these notes’ comes very close to how I would
like to teach masterclass courses or how I’d love to write books if
only I’d know how. Perhaps, over the next couple of weeks, I’ll use
this weblog again to write up a micro-course on noncommutative geometry,
some people tell me they begin to miss the mathematics on this
site.

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A for aggregates

Published January 9, 2005 by lievenlb

Let us
begin with a simple enough question : what are the points of a
non-commutative variety? Anyone? Probably you\’d say something like :
standard algebra-geometry yoga tells us that we should associate to a
non-commutative algebra $A$ on object, say $X_A$ and an arbitrary
variety is then build from \’gluing\’ such things together. Ok, but what
is $X_A$? Commutative tradition whispers $X_A=\mathbf{spec}~A$ the
[prime spectrum][1] of $A$, that is, the set of all twosided prime
ideals $P$ (that is, if $aAb \subset P$ then either $a \in P$ or $b \in
P$) and \’points\’ of $\mathbf{spec}~A$ would then correspond to
_maximal_ twosided ideals. The good news is that in this set-up, the
point-set comes equipped with a natural topology, the [Zariski
topology][2]. The bad news is that the prime spectrum is rarely
functorial in the noncommutative world. That is, if $\phi~:~A
\rightarrow B$ is an algebra morphism then $\phi^{-1}(P)$ for $P \in
\mathbf{spec}~B$ is not always a prime ideal of $A$. For example, take
$\phi$ the inclusion map $\begin{bmatrix} C[x] & C[x] \\ (x) & C[x]
\end{bmatrix} \subset \begin{bmatrix} C[x] & C[x] \\ C[x] & C[x]
\end{bmatrix}$ and $P$ the prime ideal $\begin{bmatrix} (x) & (x) \\ (x)
& (x) \end{bmatrix}$ then $P Cap \begin{bmatrix} C[x] & C[x] \\ (x) &
C[x] \end{bmatrix} = P$ but the corresponding quotient is
$\begin{bmatrix} C & C \\ 0 & C \end{bmatrix}$ which is not a prime
algebra so $\phi^{-1}(P)$ is not a prime ideal of the smaller algebra.
Failing this, let us take for $X_A$ something which obviously is
functorial and worry about topologies later. Take $X_A = \mathbf{rep}~A$
the set of all finite dimensional representations of $A$, that is
$\mathbf{rep}~A = \bigsqcup_n \mathbf{rep}_n~A$ where $\mathbf{rep}_n~A
= \{ Chi~:~A \rightarrow M_n(C)~\}$ with $Chi$ an algebra morphism. Now,
for any algebra morphism $\phi~:~A \rightarrow B$ there is an obvious
map $\mathbf{rep}~B \rightarrow \mathbf{rep}~A$ sending $Chi \mapsto Chi
Circ \phi$. Alernatively, $\mathbf{rep}_n~A$ is the set of all
$n$-dimensional left $A$-modules $M_{Chi} = C^n_{Chi}$ with $a.m =
Chi(m)m$. As such, $\mathbf{rep}~A$ is not merely a set but a
$C$-_category_, that is, all objects are $C$-vectorspaces and all
morphisms $Hom(M,N)$ are $C$-vectorspaces (the left $A$-module
morphisms). Moreover, it is an _additive_ category, that is if
$Chi,\psi$ are representations then we also have a direct sum
representation $Chi \oplus \psi$ defined by $a \mapsto \begin{bmatrix}
Chi(a) & 0 \\ 0 & \psi(a) \end{bmatrix}$. Returning at the task at
hand let us declare a _non-commutative variety_ $X$ to be (1) _an
additive_ $C$-_category_ which \’locally\’ looks like $\mathbf{rep}~A$
for some non-commutative algebra $A$ (even if we do not know at the
momemt what we mean by locally as we do not have defined a topology,
yet). Let is call objects of teh category $X$ the \’points\’ of our
variety and $X$ being additive allows us to speak of _indecomposable_
points (that is, those objects that cannot be written as a direct sum of
non-zero objects). By the local description of $X$ an indecomposable
point corresponds to an indecomposable representation of a
non-commutative algebra and as such has a local endomorphism algebra
(that is, all non-invertible endomorphisms form a twosided ideal). But
if we have this property for all indecomposable points,our category $X$
will be a Krull-Schmidt category so it is natural to impose also the
condition (2) : every point of $X$ can be decomposed uniquely into a
finite direct sum of indecomposable points. Further, as the space of
left $A$-module morphisms between two finite dimensional modules is
clearly finite dimensional we have also the following strong finiteness
condition (3) : For all points $x,y \in X$ the space of morphisms
$Hom(x,y)$ is a finite dimensional $C$-vectorspace. In their book
[Representations of finite-dimensional algebras][3], Peter Gabriel and
Andrei V. Roiter call an additive category such that all endomorphism
algebras of indecomposable objects are local algebras and such that all
morphism spaces are finite dimensional an _aggregate_. So, we have a
first tentative answer to our question **the points of a
non-commutative variety are the objects of an aggregate** Clearly, as
$\mathbf{rep}~A$ has stronger properties like being an _Abelian
category_ (that is, morphisms allow kernels and cokernels) it might also
be natural to replace \’aggregate\’ by \’Abelian Krull-Schmidt category
with finite dimensional homs\’ but if Mr. Abelian Category himself finds
the generalization to aggregates useful I\’m not going to argue about
this. Are all aggregates of the form $\mathbf{rep}~A$ or are there
other interesting examples? A motivating commutative example is : the
category of all coherent modules $Coh(Y)$ on a _projective_ variety $Y$
form an aggegate giving us a mental picture of what we might expect of a
non-commutative variety. Clearly, the above tentative answer cannot be
the full story as we haven\’t included the topological condition of
being locally of the form $\mathbf{rep}~A$ yet, but we will do that in
the next episode _B for Bricks_. [1]:
http://planetmath.org/encyclopedia/PrimeSpectrum.html [2]:
http://planetmath.org/encyclopedia/ZariskiTopology.html [3]:

1/ref=sr_1_8_1/026-3923724-4530018

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TheLibrary (version 2)

Published December 31, 2004 by lievenlb

Just in time for 2005 : a pretty good approx of what I had in mind
with TheLibrary.
The major new feature is one-page view. That
is, if you click on a bookmark or search-page link you will get a page
(as in the screenshot below) consisting of three frames. The left-bottom
frame contains the pdf file of _just_ the requested page, hence
your brwoser no longer has to download the full pdf-file to get at the
wanted page which speeds up the process. The downside is that you can no
longer scroll in neighbouring pages. To compensate for this there are
_previous page_ and _next page_ buttons in the top frame
as well as a link to the index and search page of the document.

An added bonus of this set-up is that the author of a document can
control what readers can do with these pdf-pages. For example, the pages
of 3 talks on noncommutative geometry@n
admit all features (such as content-copy, merging pages, printing etc.)
whence a determined reader can reconstruct the full pdf-document if
he/she so desires. On the other hand, the pages of version 2 can only be printed at a low
resolution and those of version 3 do not even permit this.

The bottom-right frame of the pages allow the reader
to read (and post) marginal notes wrt. the content of the document-text
(such as : extra references, errors, suggestions etc.). As always,
comments are great; obnoxious comments get deleted. Deal!
Once
again, if you like your courses and or books (on a subject from either
non-commutative geometry or non-commutative algebra) to be included in
TheLibrary email.
All scripts are adapted from the original
scripts from pdf
hacks.

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FAQ

Published December 28, 2004 by lievenlb

This is the first version of a set of general frequently asked
questions relating to _NeverEndingBooks_. An extended and updated
version of it is at all times available from the sidebar under the
heading 'FAQ'. If you have a question and/or suggestion for
these FAQs, please email and it will be answered/used in an updated
version.

What is NeverEndingBooks ?
neverendingbooks.org is a
non-profit PublishingHouse specializing in courses and books on
non-commutative algebra and/or non-commutative geometry. Our authors
have a set of notes on a subject in na&g and hope to turn it into a
book one fine day. We offer them help throughout this process, from
secure on-screen viewing & commenting of their successive versions
to the production of a genuine hardcopy version distributed worldwide.
At all times we aim to keep the costs of our books minimal and the
royalties & copyrights for our authors maximal.
NeverEndingBooks' coordinates are :

NeverEndingBooks.org

c/o Lieven Le Bruyn
Department Mathematics UA
Middelheimlaan 1
B-2020 Antwerp, Belgium

URL : www.neverendingbooks.org
ISBN-prefix : 90-809390

Why NeverEndingBooks
?
We all complain about exaggerated prices of
mathematical books from certain publishers, poor quality of editing and
refereeing offered as well as far too stringent book-contracts. Rather
than lamenting about this, NeverEndingBooks gives itself one year to
learn (and report) about the many aspects of the book-production cyclus
and to explore whether an alternative exists. If at the end of this year
we will have produced at least one book this experiment will be
considered a success. If, however, we find out that it is an impossible
task, we will explain where it all went wrong and why it is better to
stick to an established PublishingHouse and accept its terms.

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TheLibrary (demo)

Published December 23, 2004 by lievenlb

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!

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From Galois to NOG

Published December 22, 2004 by lievenlb


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.

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