Associative Composition Algebra/Split-quaternions

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

Division-binarion representation

Split-quaternions are represented by matrices of the form ${\begin{pmatrix}w&z\\z^{*}&w^{*}\end{pmatrix}}$ where w and z are division binarions. These matrices form a subalgebra of M(2,C) with determinant w w* − z z*. The group of units is obtained with non-zero determinant. When the determinant is equal to one, the subgroup is called the special unitary group and is written SU(1,1), expressing signature (1,1) with respect to the division binarions C.

Verification of the representation follows from the multiplicative properties of basis matrices:

$i={\begin{pmatrix}i&0\\0&-i\end{pmatrix}},\quad j={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad k={\begin{pmatrix}0&i\\-i&0\end{pmatrix}},\quad .$

Thus, the squares of j and k are the identity matrix, and the square of i is negative identity.

Exercise: Verify the other identities involving i, j, and k for this representation.

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

Streamlines in x-r plane, theta partial zero

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

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

In the plane x = 0, a zarmonic function is harmonic. When the partial with respect to theta is zero, a zarmonic function satisfies the wave equation in an x-r plane.

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.

Comparison to Minkowski space

Split-quaternions can be compared to Minkowski space since both are four-dimensional real vector spaces admitting transformation by circular and hyperbolic rotations. In fact, split-quaternion spacetime has signature (2,2) and has unit sphere corresponding to Lie group SU(1,1) which frequently is applied as motions of the Poincare disk model of the hyperbolic plane.

The spacetime interval $t^{2}-x^{2}-y^{2}-z^{2}$ of Minkowski space $M=\{(t,x,y,z):t,x,y,z\in \mathbb {R} \}$ has been established as the canonical empty spacetime, and it is used as the tangent space to curved spacetime with matter. The null cone in M is $K=\{(t,x,y,z):t^{2}-x^{2}-y^{2}-z^{2}=0\}$, which is used to carry light in M as cosmos, hence K is called the light cone.

Light torus

The set of null split-quaternions is N(q) = |w|² − |z|² = 0. Thus, the null elements satisfy |w| = |z| = t. In the polar form,

$w=t\exp(i\theta ),\quad z=t\exp(i\phi ).$ This null set corresponds to a parameter space

$(t,\theta ,\phi )\in \mathbb {R} _{>0}\times S^{1}\times S^{1}.$

So instead of the null cone there is a null torus as the locus of light, so it is the light torus.

Let V be the set of null vectors and (V, x) the associative magma with matrix multiplication. This magma is non-commutative and has no identity element.

The change of signature from (1,3) to (2,2) gives more dignity to light and cuts down space, for instance, conceiving longitude to be in fact a temporal variable, which is consistent with the development (1730 to 1761) of the marine chronometer by w:John Harrison.