Clifford group classification
Fred
Denote by Cl(p,q) the real Clifford algebra with p generators
squaring to +1, q to -1; and by Vl(p,q) its subset of versors,
that is products of grade-1 vectors.
What is known about the conjugacy classes of Vl(p,q)?
Real nonzero scalar factors are irrelevant, so quotiented out;
and we are interested in the number of "types", systems of conjugacy
classes parameterised by real extent parameters representing
angle, distance, etc.
As an example of the classification required, consider
kinematic (Spin-up) "rotations" represented by invertible
grade-2 versors connected to the identity, with one parameter.
Provided min(p,q) >= 2 these are of just 6 types as follows:
Hyperbolic (e.g. Moebius dilation);
Elliptic with real fixed cycles (Euclidean rotation);
Elliptic with complex fixed cycles (Lie-sphere "snake-eyes");
Parabolic semi-elliptic (Euclidean translation);
Parabolic semi-hyperbolic (Laguerre offset);
Parabolic (Laguerre "floorboard").
Fred
Fred
I've uploaded a file with some relevant background.
It was intended to be informal; plaintext; brief; and
stimulating. Enlightenment with regard to the contents
would be appreciated. Fred
Fred lunnon
> > ...
> > In the future just send me an email directly if you want an uploaded
> > file deleted.
> > ...
One more time, please Peeter --- I've uploaded <lurker2.txt> again,
but it seems to have stabilised now!
[In any case, if I work on this approach any more, I'll typeset it in LaTeX
properly. I had hoped to catch people's interest with a stripped-down
account, but it doesn't seem to have been too successful so far ...]
On a different matter --- I have some GA software and demonstrations
in Java and Maple, which I had wondered about making available on
geometric_algebra. These probably should go into a separate directory
--- there are currently around 16 files totalling 0.5Mbyte of source code
and documentation.
How would you feel about this? Fred Lunnon
Peeter Joot
Once again, note that stuff you send to the group goes to all subscribers. Send maintainance related stuff to me directly at:
peete...@gmail.com
Personally I found it too much work to read that ascii doc you uploaded, and until it is formatted in latex I wouldn't try. Perhaps lack of comment from others is due to the same reason?
As for the software, I have some too, and I suggest that a public git repository on github is more appropriate. What version control software are you currently using? I have some symbolic GA code based on STL and gaigen, and wouldn't object to moving that to a common repository for misc GA software.
Peeter
ehitzer
There is in fact new work on the subject in the book by
H. Li, Invariant Algebras and Geometric Reasoning, World Scientific,
2008.
Especially chapter 6.2 on Versor compression.
With kind regards,
Eckhard
Fred
Eckhard,
Thanks for this reference, which certainly looks relevant ---
tho' I have so far alas failed to actually get sight of it. However,
a few clues picked up elsewhere suggest that "versor
compression" concerns the factorisation of a grade-k versor
into k vectors. Presumably, the method used is essentially
that of the (constructive) proof of Cartan-Dieudonne ---
take a vector at random, transform it by the versor, add
(bisect) the two to get a vector which factors the versor,
then carry on inductively ... can you confirm this?
The classification problem depends on a rather different
algorithm --- unique factorisation of an even versor as the
product of orthogonal (bivector) rotations. Li's book doesn't
seem to discuss this, as far as I can judge --- chapter 7 has
sections on rotations, but nothing about higher grades.
Once again, can you comment?
WFL