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Tag: geometry

pdfsync

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;

One Comment

why nag? (3)

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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writing

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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