Yes, but they differ in how we associate (some compactification of) R^n with (some quotient of) S^n. There is, however, a rather nice fact: Let G be the group of projective transformations of R^(n+1) which preserve S^n. Let H be the group of conformal transformations of R^(n+1) which preserve S^n. Then G and H are isomorphic, and we can choose the isomorphism such that they act on S^n in the same way.
Sent: Friday, August 29, 2014 at 12:31 AM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Stupid question about geometrical transformations
I don't think incompatible compactifications is a problem.
The conformal group on n dimensions is generated by inversions in (n-1)-spheres. It naturally lives on the n-sphere S^n.
The projective group on n+1 dimensions naturally lives on projective space P^n, by the action of linear transformations on lines through the origin in R^(n+1).
But by just looking at the action on rays instead of lines (or equivalently, taking the double cover of P^n), we get the projective group acting on S^n also.
So, both the projective group and the conformal group act naturally on S^n.
Is Adam's answer compatible with this point of view? I'm not sure I understand it.
--Dan
On Aug 28, 2014, at 12:34 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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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