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connected

Published November 1, 2004 by lievenlb

If this message gets posted it will mean that I finally
succeeded in connecting LeTravers to the rest of the world…
Clearly not via cable but using good old dial-in. I don't think
I'll ever see cable appearing here.

Electricity made it
appearance here only 10 years ago (and is an end-of-network setup
meaning that if two people on the mountain use a microwave, all lights
are dimmed…) and since 5 years one can reach us by telephone.

Since then I've been trying to get email working using
all sorts of (Belgian) dial-in adresses but nothing worked, the modem
didn't seem to be working. It turned out that in France you first
have to buy a special socket for the telephone outlet (costs 50FF)
which our neighbors promised to provide by the next time we came
along.

So, next time expectations were high and sure enough
I could hear the typical modem-noises until they got into an infinite
loop without ever making the connection.

Some people were
luckier but then they used a Windows-clone and even mimicking their
connection on a Mac didn't work. For some mysterious reason it
seemed that Macintosh computers (or at least their modems) were
incompatible with FranceTelecom.

Last week I did try
another option : I got a webpage with all free internet providers and
applied for a username-password with two of them (FreeFrance and
Tiscali). FreeFrance promised to send a package with the post whereas
Tiscali immediately replied with a dial-in nummer, username and
allowed me to set up my own password.

So, after driving
1000km (half of which in the pouring rain) and enjoying a glass of
rose outside in the setting sun (picture) I tried the Tiscali
connection without too much hope, but I think it works.

It
was a beautiful sunny afternoon (it seems it has been raining here
more or less continuously for the last three weeks) but at sunset the
clouds were rather threatening and sure enough the following day
(sunday) we spend the day within rain clouds.

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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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simple groups

Published October 24, 2004 by lievenlb

I
found an old copy (Vol 2 Number 4 1980) of the The Mathematical Intelligencer with on its front
cover the list of the 26 _known_ sporadic groups together with a
starred added in proof saying

  • added in
    proof … the classification of finite simple groups is complete.
    there are no other sporadic groups.

(click on the left picture to see a larger scanned image). In it is a
beautiful paper by John Conway “Monsters and moonshine” on the
classification project. Along the way he describes the simplest
non-trivial simple group $A_5 $ as the icosahedral group. as well as
other interpretations as Lie groups over finite fields. He also gives a
nice introduction to representation theory and the properties of the
character table allowing to reconstruct $A_5 $ only knowing that there
must be a simple group of order 60.
A more technical account
of the classification project (sketching the main steps in precise
formulations) can be found online in the paper by Ron Solomon On finite simple
groups and their classification. In addition to the posts by John Baez mentioned
in this
post he has a few more columns on Platonic solids and their relation to Lie
algebras, continued here.

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