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Tag: geometry

version pi

Published November 10, 2004 by lievenlb

Now
that versions 2 and 3 of my abandoned book project
noncommutative~geometry@n are being referenced (as suggested) as
“forgotten book” (see for example Michel's latest paper) it is
perhaps time to consider writing version $\\pi$. I haven't made up
my mind what to include in this version so if you had a go at these
versions (available no longer)
and have suggestions, please leave a comment. An housekeeping-note :
this blog is flooded with link-spammers recently so I did remove the
automatic posting of comments. I use the strategy proposed by Angsuman to combat
them. This sometimes means that I overlook a comment (this morning I
discovered a lost comment while cleaning up the spam-comments, sorry!)
but it is the only way to keep this blog poker-casino-sex-etc free. It
goes without saying that any relevant comment (positive or negative)
will be approved as soon as I spot it.

At the moment I
haven't the energy to start the writing phase yet, but I am slowly
preparing things

  • Emptied the big antique table upstairs
    to have plenty of place to put things.
  • Got myself a laser
    printer and put it into our home-network using AirportExpress which
    allows to turn any USB-printer into a network-printer.
  • Downloaded the Springer Verlag Book Stylefiles svmono.zip. This
    does not mean that I will submit it there (in fact, I promised at least
    one series-editor to send him a new version first) but these days I
    cannot bring myself to use AMS-stylefiles.
  • Accepted an
    invitation to give a master-course on noncommutative geometry in Granada in 2005 which, combined with
    the master-class here in Antwerp next semester may just be enough
    motivation to rewrite notes.
  • Bought all four volumes of the
    reprinted Winning Ways for your
    Mathematical Plays     as inspiration for fancy terminology and notation
    (yes, it will be version $\\pi$ and _not_ version $e$).
  • etc.
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padlock returns

Published November 5, 2004 by lievenlb

A couple of months ago I spend some time modifying the WordPress ViewLevel
plugin slightly to include in this blog. At the time, the idea was
to restrict the readership of certain posts (such as info meant for
master-class students etc.). In the sidebar these posts are prepended by
a padlock sign (together with the appropriate view-level). In the main
window these posts do _not_ show up unless you are logged in and
have the fitting view-level.
I hope that this tool may also prove
useful to combat spam-comments. Ideally, a weblog should be configured
to accept any comments but if you have to remove a 100 or more link-spam
'comments' each morning to keep your blog poker-free you have to
play defensive. Unfortunately, WordPress is not very good at it. Sure,
one can opt to put all comments on hold, awaiting moderation but (1)
this is unpleasant for genuine comments and (2) one still has to remove
all spam-comments manually from the moderation-queue. In the end, I had
to close all posts for comments to be spared from poker-online and
texas-online rubbish.
However, I appreciate comments and
suggestions especially at a time when this weblog is changing. So, if
you are working in either non-commutative algebra or non-commutative
geometry and want to give your suggestions, please get yourself a login.
I know, I know, it is a hassle with all those nonsense passwords but if
you are accessing this weblog from just one computer you only have to
remember it once (I forgot my own password but can still post
here…). I will then raise your ViewLevel from the default 0 value
to at least 1 so that you can read and comment the padlocked posts. If
you then want to make a comment on other posts, please use a nearby
padlocked post.
Today, I ask for suggestions for a good LaTeX
book-style. At the moment my favourite is the CTAN
thesis-package but surely there are better packages out there!

again : this idea came to nothing!

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double Poisson algebras

Published October 26, 2004 by lievenlb

This morning,
Michel Van den Bergh posted an interesting paper on the arXiv
entitled Double
Poisson Algebras. His main motivation was the construction of a
natural Poisson structure on quotient varieties of representations of
deformed multiplicative preprojective algebras (introduced by
Crawley-Boevey and Shaw in Multiplicative
preprojective algebras, middle convolution and the Deligne-Simpson
problem) which he achieves by extending his double Poisson structure
on the path algebra of the quiver to the 'obvious' universal
localization, that is the one by inverting all $1+aa^{\star} $ for $a $ an
arrow and $a^{\star} $ its double (the one in the other direction).
For me the more interesting fact of this paper is that his double
bracket on the path algebra of a double quiver gives finer information
than the _necklace Lie algebra_ as defined in my (old) paper with Raf
Bocklandt Necklace
Lie algebras and noncommutative symplectic geometry. I will
certainly come back to this later when I have more energy but just to
wet your appetite let me point out that Michel calls a _double bracket_
on an algebra $A $ a bilinear map
$\{ \{ -,- \} \}~:~A \times A
\rightarrow A \otimes A $
which is a derivation in the _second_
argument (for the outer bimodulke structure on $A $) and satisfies
$\{ \{ a,b \} \} = – \{ \{ b,a \} \}^o $ with $~(u \otimes v)^0 = v
\otimes u $
Given such a double bracket one can define an ordinary
bracket (using standard Hopf-algebra notation)
$\{ a,b \} = \sum
\{ \{ a,b \} \}_{(1)} \{ \{ a,b \} \}_{(2)} $
which makes $A $ into
a Loday
algebra and induces a Lie algebra structure on $A/[A,A] $. He then
goes on to define such a double bracket on the path algebra of a double
quiver in such a way that the associated Lie structure above is the
necklace Lie algebra.

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