On reflection I agree with this. In fact, the algebra I favour for Euclidean geometry in 3-space is the Clifford Cl(3, 0, 1) , which puzzled me initially by representing a finite point in two ways --- the "doppelganger" arising from the polar opposite on the Riemann sphere. Adam shows how to construct algebraic curves of arbitrary degree via sequences of projectivities and Moebius transformations acting on an initial line. I suppose that strictly speaking it doesn't immediately follow that one can construct sufficiently many to force an unbounded dimension ... More pondering may be required? WFL On 8/29/14, Dan Asimov <dasimov@earthlink.net> wrote:
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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