Associative Composition Algebra/Split-quaternions

There are at least three portals leading to split quaternions: the dihedral group of a square, matrix products in M(2,R), and the modified Cayley-Dickson construction. The work of Max Zorn on split octonions showed the necessity of including split real AC algebras in the aufbau of the category.

The imaginary units v, such that v ² = −1, lie on a two-sheeted hyperboloid in split quaternions

The development through the dihedral group was started with a lemma in the Introduction, and is completed with exercises below.

Or one can start with a basis {1, i, j, k} taken from M(2,R), where the identity matrix is one, ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ is j, ${\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$ is i, and ${\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$ is k. Some practice with matrix multiplication shows they are anticommutative like division quaternions, but some products differ:

j² = +1 = k²,   j k = − i .

Then the real AC algebra of split-quaternions uses coefficients w, x, y, z ∈ R to express an element, its conjugate, and the quadratic form N:

$q=w+xi+yj+zk,\quad q^{*}=w-xi-yj-zk,\quad N(q)=w^{2}+x^{2}-y^{2}-z^{2}.$

Exercises

1. What are the involutions on a square ?

2. As reflections, what is the angle of incidence of the axes of reflection ?

3. The composition of these reflections has what angle of rotation ?

Computations

Insight into the structure and dynamics of split-quaternions is available through elementary computational exercises. These exercises use j² = +1 = k² and jk = −i, contrary to the quaternion group, which is expressed with the same letters i, j, k, but which here refer to the dihedral group of a square instead.

1. For r = j cos θ + k sin θ, show that r² = +1 = −r r*.
2. Compute ir .
3. Recall that <q, t> = (q t* + t q*)/2. Show <q, t> = real part of q t*
4. Definition: q and t are orthogonal when <q, t> = 0.
5. Show that for any theta, r and ir are orthogonal.
6. Let p = i sinh a + r cosh a. Show that p² = +1 for any a and r.
7. Let v = i cosh a + r sinh a. Show that v² = −1.
8. For a given a and r, show that p and v are orthogonal.
9. Let m = p exp(bp) = sinh b + p cosh b. Show that m m* = −1.
10. Let w = exp(bp) = cosh b + p sinh b. Show that m is orthogonal to w.
11. Show that m is orthogonal to v.
12. For any θ, a, and b defining r, p, w, v, and m, the set {m, w, v, ir} is an orthonormal basis.
13. If u is a unit, <qu, tu> = uu* <q, t>.

Vortex

Cyclone Catarina, March 26, 2004

The natural phenomena of hurricanes, tornados, waterspouts, and dust devils illustrate the vortex motion. They have in common a circular motion and upwelling in the center. The split-quaternions contribute to representation of such a velocity field when a certain differential operator is introduced.

Toy vortex

Fluid responds to differential rotation of cylinders

Concentric rotating cylinders have been used to study fluid motion in response to differential rotation of the cylinders. With a translucent outer cylinder, and bits of pepper to exhibit the fluid motion, there are various states of the system. With a small differential between rates of rotation of the inner and outer cylinder the state of laminar flow occurs. Each increment of radius is a lamina rotating in a circle. With a somewhat higher differential a secondary motion appears in the various meridians of the model. This motion corresponds to the upwelling seen in the natural phenomena. A third state of motion in the model has waves forming in the circulating mass. The principal investigators in this experimental setup were w:Arnulph Mallock, w:Maurice Couette, and w:G. I. Taylor.

Zabla

The differential operator "nabla" or "del" is $\nabla =i{\frac {\partial }{\partial x}}+j{\frac {\partial }{\partial y}}+k{\frac {\partial }{\partial z}}$ . A velocity potential is a function $f:R^{3}\to R$.

When nabla is applied to a velocity potential it gives a velocity field, a vector-valued function.

Zabla is the same operator as in quaternion case, but there is a difference when it is squared:

$\nabla ^{2}=-{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}$ .

Zarmonic functions

A harmonic function f has $\nabla ^{2}f=0$ with the nabla convention, while a function will be called zarmonic when the zabla convention holds, as here with split-quaternions where the square of j or k is opposite the square of i.

Let $y^{2}+z^{2}=r^{2}$ with the y-z plane described in polar coordinates (r, θ). Then

${\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}={\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}.$

The three states of motion in the toy vortex arise as terms differ from zero. The laminar flow, such as f(r) = log r, corresponds to partials with respect to x and θ as zero. The meridian flow starts when the partial with respect to x departs from zero. The wavy characteristic show that partial with respect to theta is non-zero.