Coquaternion

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In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. Like the 1843 quaternions of Hamilton, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, coquaternions may be zero divisors, idempotent, or nilpotent.

The set ${1, i, j, k}$ forms a basis. The coquaternion products of these elements are

$i j = k = -j i, ~ ~ j k = -i = -k j, ~ ~ k i = j = -i k$

$i^2 = -1, ~ ~ j^2 = +1, ~ ~ k^2 = +1$ .

This set can be identified as the basis elements of either the Clifford algebra Cℓ_1,1(R), with {1, e₁=i, e₂=j, e₁e₂=k}; or the algebra Cℓ_2,0(R), with {1, e₁=j, e₂=k, e₁e₂=i}.

As the latter relationship implies, with these products the set ${1, i, j, k, - 1, - i, - j, - k}$ is isomorphic to the dihedral group of a square.

A coquaternion

$q~= w + x i + y j + z k$

has conjugate

$q^* ~= w - x i - y j - z k$

and multiplicative modulus

$qq^* ~= w^2 + x^2 - y^2 - z^2$ .

When the modulus is non-zero, then q has a multiplicative inverse.

$U = \{q : qq^* \ne 0 \}$

is the set of units. The set P of all coquaternions forms a ring (P, +, •) with group of units (U, •).

Let

$q = w + x i + y j + z k, ~ ~ u = w + x i, ~ ~ v = y + z i$

where u and v are ordinary complex numbers. Then the complex matrix

$\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix}$ ,

where $u * = w - x i$ and $v * = y - z i$ (complex conjugates of u and v), represents q in the ring of matrices in the sense that multiplication of coquaternions behaves the same way as the matrix multiplication. For example, the determinant of this matrix $u u * - v v * = q q *$ ; the appearance of this minus sign where there is a plus in H leads to the alternative name split-quaternion for a coquaternion. Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra.

1 Profile
2 Pan-orthogonality
3 Counter-sphere geometry
4 Isomorphism with M(2,R)
5 Application to kinematics
6 Historical notes and references
7 See also

[edit] Profile

Let

r(θ) = j cos θ + k sin θ (here θ is as fundamental as azimuth)

p(a, r) = i sinh a + r cosh a

v(a, r) = i cosh a + r sinh a

E = { r ∈ P: r = r(θ), 0 ≤ θ < 2 π}

J = {p(a, r) ∈ P: a ∈ R, r ∈ E} catenoid

I = {v(a, r) ∈ P: a ∈ R, r ∈ E} hyperboloid of two sheets

Now it is easy to verify that

{q ∈ P: q² = + 1} = J ∪ {1, -1}

and that

{q ∈ P: q² = -1} = I.

These set equalities mean that when p ∈ J then the plane

{x + yp: x, y ∈ R} = D_p

is a subring of P that is isomorphic to the plane of split-complex numbers just as when v is in I then

{x + yv: x, y ∈ R} = C_v

is a planar subring of P that is isomorphic to the ordinary complex plane C.

Note that for every r ∈ E, (r + i)² = 0 = (r - i)² so that r + i and r - i are nilpotents. The plane N = {x + y(r + i): x, y ∈ R} is a subring of P that is isomorphic to the dual numbers. Since every coquaternion must lie in a D_p, a C_v, or an N plane, these planes profile P. For example, the unit sphere

SU(1, 1) = {q ∈ P: qq* = 1}

consists of the "unit circles" in the constituent planes of P. In D_p this is an hyperbola, in N the unit circle is a pair of parallel lines, while in C_v it is indeed a circle (though it appears elliptical due to v-stretching).These ellipse/circles found in each C_v are like the illusion of the Rubin vase which "presents the viewer with a mental choice of two interpretations, each of which is valid".

[edit] Pan-orthogonality

When coquaternion $q = w + x i + y j + z k$ , then the real part of q is w.
Definition: For non-zero coquaternions q and t we write q ⊥ t when the real part of the product $q t *$ is zero.

For every v ∈ I, if q, t ∈ C_v, then q ⊥ t means the rays from 0 to q and t are perpendicular.
For every p ∈ J, if q, t ∈ D_p, then q ⊥ t means these two points are hyperbolic-orthogonal.
For every r ∈ E and every a ∈ R, p = p(a, r) and v = v(a, r) satisfy p ⊥ v.
If u is a unit in the coquaternion ring, then q ⊥ t implies qu ⊥ tu.

Proof:

(q u)(t u) * = (u u *) q t *

follows from

(t u) * = u * t *

, a fact based on anti-commutativity of vectors.

[edit] Counter-sphere geometry

Take $m = x + y i + z r$ where $r~= j \cos \theta + k \sin \theta$ . Fix theta (θ) and suppose

m m * = - 1 = x 2 + y 2 - z 2

Since points on the counter-sphere must line on a counter-circle in some plane D_p ⊂ P, m can be written, for some p ∈ J

$m~= p \exp{(bp)} = \sinh b + p \cosh b = \sinh b + i \sinh a~\cosh b + r \cosh a~\cosh b$ .

Let φ be the angle between the hyperbolas from r to p and m. This angle can be viewed, in the plane tangent to the counter-sphere at r, by projection:

$\tan \phi = \frac{x}{y} = \frac{\sinh b}{\sinh a ~\cosh b} = \frac{\tanh b}{\sinh a}$ .

As b gets large, tanh b nears one. Then tan φ = 1/sinh a . This appearance of the angle of parallelism in a meridian θ inclines one to expect to see the counter- sphere unfold as the manifold S¹ × H² where H² is the hyperbolic plane.

[edit] Isomorphism with M(2,R)

Besides the complex matrix representation given above, another linear representation associates coquaternions with Real matrices (2 x 2). Note first the product $\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}$ and that the square of each factor on the left is the identity matrix, while the square of the right hand side is the negative of the identity matrix. Furthermore, note that these three matrices, together with the identity matrix, form a basis for M(2,R). One can make the matrix product above correspond to j k = −i in the coquaternion ring. Then for an arbitrary matrix there is the Bijection

$\begin{pmatrix} a & c \\ b & d\end{pmatrix} \leftrightarrow q = [(a+d) + (c-b)i + (b+c)j + (a-d)k]/2,$

which is in fact a ring isomorphism. Furthermore, computing squares of components and gathering terms shows that q q * = ad − bc , which is the determinant of the matrix. Consequently there is a group isomorphism between the unit sphere of coquaternions and SL(2,R) = {g ∈ M(2,R) : det g = 1 }, and hence SU(1,1) ≈ SL(2,R).

[edit] Application to kinematics

By using the foundations given above, one can show that the mapping

q → u⁻¹qu

is an ordinary or hyperbolic rotation according as

u = exp(av), v ∈ I or u = exp(ap), p ∈ J.

These mappings are projectivities in the inversive ring geometry of coquaternions. The collection of these mappings bears some relation to the Lorentz group since it is also composed of ordinary and hyperbolic rotations. Among the peculiarities of this approach to relativistic kinematic is the anisotropic profile, say as compared to Hyperbolic quaternions.

Reticence to use coquaternions for kinematic models may stem from the (2, 2) signature when spacetime is presumed to have signature (1, 3) or (3, 1). Nevertheless, a transparently relativistic kinematics appears when a point of the counter-sphere is used to represent an inertial frame of reference. Indeed, if $t t * = - 1$ , then there is a p ∈ J such that t ∈ D_p, and an a ∈ R such that t = p exp(ap). Then if u = exp(ap) and s = ir, the set {t, u, v, s} is a pan-orthogonal basis stemming from t, and the orthogonalities persist through applications of the ordinary or hyperbolic rotations.

[edit] Historical notes and references

The coquaternions were initially identified and named in the London-Edinburgh-Dublin Philosophical Magazine, series 3, volume 35, pp. 434,5 in 1849 by James Cockle under the title "On Systems of Algebra involving more than one Imaginary". At the 1900 Paris meeting of the International Congress of Mathematicians, Alexander MacFarlane called the algebra the exspherical quaternion system as he described its profile. MacFarlane examined a differential element of the submanifold {q ∈ P: $q q * = - 1$ } (the counter-sphere).

The sphere itself was considered in German by Hans Beck in 1910 (Transactions of the American Mathematical Society, v. 28; e.g. the dihedral group appears on page 419). In 1942 and 1947 there were two brief mentions of the coquaternion structure in the Annals of Mathematics: