It's infinite-dimensional. We can certainly represent these things as maps of the form: (x, y, z) --> (f(x,y,z), g(x,y,z), h(x,y,z)) [for two dimensions; generalise as appropriate] where f, g and h are homogeneous rational functions, where deg(numerator) = deg(denominator) + 1.
Sent: Thursday, August 28, 2014 at 8:34 PM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Stupid question about geometrical transformations
Oops --- should read (n+1)^2-1 + (n+2)(n+1)/2 - (n+1)n/2 - 1 = n^2 + 3n .
One immediate difficulty is incompatible compactifications --- a hyperplane versus a single point at infinity.
WFL
On 8/28/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
What is generated by the union of projective and conformal (Moebius) groups?
Since these two intersect in similarities, the super-group in n-space has dimension at least (n^2-1) + (n+2)(n+1)/2 - (n+1)n/2 - 1 = n^2 + n - 1 ; just how big is it?
How should such transformations be represented for computational purposes?
Why don't I know the answers to these apparently obvious questions? [Uh, maybe don't answer that one right now ...]
Physicists have previously devoted some thought to this matter: in particular, a promising paper by Wolfgang Bertram (2001) at http://www.emis.de/journals/AG/2-4/2_329.pdf launches into discussing "Jordan functors", which will however surely cost this innocent much gruesome effort to decode.
[Pascual Jordan certainly seems put himself about, despite which I don't recall ever having encountered him before this week.]
Fred Lunnon
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