non-commutative – Page 7 – neverendingbooks

GMD

Published April 14, 2005 by lieven

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 isn√ït in the right place (close all open
loops)
get rid of the stuff that isn√ït yours or you don√ït
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.

pdfsync

Published April 13, 2005 by lieven

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;

why nag? (3)

Published March 29, 2005 by lieven

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!

Tag: non-commutative

GMD

pdfsync

why nag? (3)