On 12/16/09, Dan Asimov <dasimov@earthlink.net> wrote:
Fred wrote:
<< . . . Unfortunately, quaternions could be modelled by the Clifford algebra Cl(0,2); but it's a very bad idea because their symmetry is then lost. . . .
What does "could be modelled by" mean, or for that matter, "is modelled by" ?
Careless wording --- I meant that the rings involved are isomorphic. I'll denote this \H ~= Cl(0,2) --- unless anybody can suggest a better-established alternative. It must be admitted that more precise nomenclature is called for here --- for example, as earlier observed \H ~= Cl(3,0)^0 ~= Cl(0,3)^0 (even-grade subalgebra), with dimension 4. In all cases above, the "versor" semigroup (preserving vectors) happens to equal the whole "multor" (Clifford) algebra. But which in general the inclusion is proper, causing my later muddle: for example, the complex bi-quaternions \H (x) \C ~= Cl(3,0), are isomorphic to the multors with dimension 8, rather than to its versors (even and odd), with dimension only 4.
(And does "their" refer to the quaternions or to Cl(0,2) ?)
The quaternions have a natural order-3 automorphism i -> j -> k -> i, which Cl(0,2) lacks. This causes both computational inconvenience and conceptual confusion, when neophytes attmept to impose a symmetry which doesn't exist. [Notation for versor set might be warrented here --- "Vl(p, q)" maybe --- the "Clifford group" of invertible versors is often denoted by uppercase gamma. I didn't initially consider discussing quaternions in this little screed; but maybe a separate section along these lines, employing them as a relatively familiar example, does now seem a good idea after all.] It's perhaps worth emphasising why we should deal with versors rather than multors --- particularly since established references are fulsomely specific about the algebraic structure of the latter, much less so the former. The essential point is that reversion delivers a Clifford-product pseudo-inverse --- permitting efficient transformation via conjugation --- only for a versor; for a general multor, computing the inverse is much harder. [The situation is analogous to transposition of an orthogonal matrix, compared to general matrix inversion.] Fred Lunnon