On 12/15/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 12/15/09, Fred lunnon <fred.lunnon@gmail.com> 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.
A much better model is the even subalgebra of the complex biquaternions Cl(3,0): with generators \x,\y,\z, we have i = \z\y, j = \x\z, k = \y\x.
Blooper: complex biquaternions have dimension 8 over \R; the even part of Cl(3,0) had dimension 4, and has index 2 in the full (versor) group (which comprises the odd part as well). So the latter has dimension 4 as well --- duh!
Note that these are bivectors (grade 2), representing rotations in space --- as opposed to vectors (grade 1), which generally represent prime reflections (that is, in "hyperplanes").
There's another wrinkle involved here which I hadn't appreciated before: embedding \H -> Cl(0,2) maps the quaternions onto the whole algebra, not just its versors. This means that none of the machinery I'm discussing here is actually applicable --- another good reason for binning it [the duff embedding, not the Clifford algebra!]
This confusion over quaternions is just one of many traps for the unwary ...
Quite so ... WFL