[math-fun] Lie-sphere lurker
Feedback invited: errors, obscurities, typos, "stupid questions" and all. Text key: "^" for superscript, "_" for subscript, "\" for boldface. First instalment is a skeletal summary introducing 1. Clifford Algebras ____________________ A (nondegenerate) Clifford algebra Cl(p,q) over some "scalar" field --- usually the real numbers \R, occasionally the complex numbers \C --- extends it to polynomials in a set of transcendental "generators" subject to the usual arithmetic laws of polynomial rings, except that: multiplication of generators is anti-commutative; and generators square to +1 (p of them) or to -1 (q of them). Scalars are embedded in the algebra in the natural fashion, as elements of grade 0 (constant polynomials) or -1 (zero); vectors of dimension p+q as elements of grade 1 (linear polynomials). A "versor" is any element which conjugates vectors to vectors: if F is linear, then so is G = A^+ F A. Here A^+ denotes "reversion" of A, which acts as an inverse to the extent that the "magnitude" ||A|| == A^+ A is scalar. For most purposes, nonzero scalar factors are irrelevant; zero magnitude on the other hand indicates a non-invertible projection. It's fortunate that the reversion is easily computed: if the terms of A are broken down into subsums <A>_k by grade (number of generators) A = <A>_0 + <A>_1 + <A>_2 + <A>_3 + <A>_4 + ... then A^+ == <A>_0 + <A>_1 - <A>_2 - <A>_3 + <A>_4 + ... Finally, A is invertible just when ||A|| is nonzero. WFL
On 12/15/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... First instalment is a skeletal summary introducing
1. Clifford Algebras ____________________ ...
Andy has queried how quaternions fit into this framework. 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. 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"). This confusion over quaternions is just one of many traps for the unwary ... Fred
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
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Fred lunnon