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.
neverendingbooks Posts
This week I finished the first phase of my home-network plans : from
every computer one can stream iTunes-music files to be played in
the living room, and from every computer one can stream iPhoto or
iMovie-files to the TV-set. Both solutions involve new hardware
and as it usually is with hardware : there are elegant or clumsy
solutions. The photo-video solution is rather clumsy so I’ll postpone
it until later. The audio-solution on the other hand is extremely
elegant : I bought a squeezebox from slimdevices.
It is extremely small (22cmx5cmx12cm) so you can place it virtually
everywhere, it can be connected to your network either wireless or by
ethernet and it has several alternatives to connect to your speakers or
hifi-set : a headphone mini-jack (which is at the moment what I use to
connect it to a pair of powered speakers) but I can always upgrade my
listening experience using : analog audio RCA, digital optical or
digital coax. The service it provides is excellent, all information is
available from their website (they do not ship a CD but you can download
the latest version of slimserver from the website (available for all
platforms)) and they have several forums among which a rather enthusiastic
users-forum (no surprise to me). You get it running in no time. First
download slimserver and install it on the machine(s) containing music
you want to stream over the network. What it does is to add one extra
item to your SystemPreferences and clicking it you can start the
SlimServer. Then, power up the squeezebox and
follow the messages on the display. I choose to connect to a wireless
network, it then detects the names of the possible networks and asks you
to choose one, asks what type of encryption it uses (for Airport b take
64 bit, for g take 128 bit encryption). The most difficult part is to
type in the wireless password as it wants the hexadecimal
equivalent of your pass phrase. To find it, open up Airport Admin
Utility, log in to the network you want and on the screen you get
look for the password icon, clicking it will give you the
hexadecimal WEP-key. If you are familiar with SMS you’ll find it easier
than me to type this in to the squeezebox
(use the number keys to simulate a keyboard). After this it will connect
to your network and get the IP-address of the SlimServer (the computer
on which you started the server) and you have access to its
iTunes-library. Two caveats : make sure you use the MP3-option to
get CD’s into iTunes (the default setting of Preferences/import
is AAC not MP3 (btw. slimdevices now claim that you may also use
AAC-files, I didn’t check this out yet but have no doubts it will
work). The second is that the display screen is rather small to browse
the library if you are used to iTunes’ window. A neat way around it is
to use a webbrowser on ANY computer in your network (for example the
iBook on your lap). Just fire up Safari and go to the
Rendezvous-window (it took me some time to figure out what they
meant by it : in Safari go to the Bookmarks pulldown menu, then
choose Show all bookmarks and on this page you will find a
Rendezvous-tab, click it and one of the bookmarks will be
SlimServer and by clicking on it you have web-access to control
your squeezebox. Very neat, this Rendezvous
awareness and a sharp contrast to the clumsy photo-video solution. So,
if you want to free your music and hear it via hifi-standards rather
than via computer-soundcards go and buy as many squeezebox sets as you will need to fill your house with
music!