Split quaternion

Matrix representations

Let

q = w + xi + yj + zk,

and consider u = w + xi, and v = y + zi as ordinary complex numbers with complex conjugates denoted by u* = w − xi, v* = y − zi. Then the complex matrix

( u v v ∗ u ∗ ) ,

represents q in the ring of matrices, i.e. the multiplication of split-quaternions behaves the same way as the matrix multiplication. For example, the determinant of this matrix is

uu* − vv* = qq*.

The appearance of the minus sign, where there is a plus in H, distinguishes coquaternions from quaternions. The use of the split-quaternions of norm one (qq* = 1) for hyperbolic motions of the Poincaré disk model of hyperbolic geometry is one of the great utilities of the algebra.

Besides the complex matrix representation, another linear representation associates coquaternions with 2 × 2 real matrices. This isomorphism can be made explicit as follows: Note first the product

( 0 1 1 0 ) ( 1 0 0 − 1 ) = ( 0 − 1 1 0 )

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 jk = −i in the coquaternion ring. Then for an arbitrary matrix there is the bijection

( a c b d ) ↔ 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 qq* = ad − bc, which is the determinant of the matrix. Consequently there is a group isomorphism between the unit quasi-sphere of coquaternions and SL(2, R) = {g ∈ M(2, R) : det g = 1}, and hence also with SU(1, 1): the latter can be seen in the complex representation above.

For instance, see Karzel and Kist for the hyperbolic motion group representation with 2 × 2 real matrices.

In both of these linear representations the norm is given by the determinant function. Since the determinant is a multiplicative mapping, the norm of the product of two coquaternions is equal to the product of the two separate norms. Thus coquaternions form a composition algebra. As an algebra over the field of real numbers, it is one of only seven such algebras.

Generation from split-complex numbers

Kevin McCrimmon has shown how all composition algebras can be constructed after the manner promulgated by L. E. Dickson and Adrian Albert for the division algebras C, H, and O. Indeed, he presents the multiplication rule

( a , b ) ( c , d )   =   ( a c + d ∗ b ,   d a + b c ∗ )

to be used when producing the doubled product in the real-split cases. As before, the doubled conjugate ( a , b ) ∗   =   ( a ∗ , − b ) , so that

N ( a , b )   =   ( a , b ) ( a , b ) ∗   =   ( a a ∗ − b b ∗ , 0 ) .

If a and b are split-complex numbers and split-quaternion q   =   ( a , b )   =   ( ( w + z j ) , ( y + x j ) ) ,

then N ( q ) = a a ∗ − b b ∗ = w 2 − z 2 − ( y 2 − x 2 ) = w 2 + x 2 − y 2 − z 2 .

Profile

The subalgebras of P may be seen by first noting the nature of the subspace {zi + xj + yk : x, y, z ∈ R}. Let

r(θ) = j cos(θ) + k sin(θ)

The parameters z and r(θ) are the basis of a cylindrical coordinate system in the subspace. Parameter θ denotes azimuth. Next let a denote any real number and consider the coquaternions

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

These are the equilateral-hyperboloidal coordinates described by Alexander Macfarlane and Carmody.

Next, form three foundational sets in the vector-subspace of the ring:

E = {r ∈ P: r = r(θ), 0 ≤ θ < 2π}J = {p(a, r) ∈ P: a ∈ R, r ∈ E}, hyperboloid of one sheetI = {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 quasi-sphere

SU(1, 1) = {q ∈ P: qq* = 1}

consists of the "unit circles" in the constituent planes of P: In D_p it is a unit 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".

Pan-orthogonality

When coquaternion q = w + xi + yj + zk, then the scalar part of q is w.

Definition. For non-zero coquaternions q and t we write q ⊥ t when the scalar 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: (qu)(tu)* = (uu*)q(t*) follows from (tu)* = u*t*, which can be established using the anticommutativity property of vector cross products.

Counter-sphere geometry

The quadratic form qq* is positive definite on the planes C_v and N. Consider the counter-sphere {q: qq* = −1}.

Take m = x + yi + zr where r = j cos(θ) + k sin(θ). Fix θ and suppose

mm* = −1 = x² + y² − z².

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

m   = p exp ⁡ ( b p ) = 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 ⁡ ϕ = x y = sinh ⁡ b sinh ⁡ a   cosh ⁡ b = tanh ⁡ b sinh ⁡ a . Then lim b → ∞ tan ⁡ ϕ = 1 sinh ⁡ a ,

as in the expression of angle of parallelism in the hyperbolic plane H² . The parameter θ determining the meridian varies over the S¹. Thus the counter-sphere appears as the manifold S¹ × H².

Application to kinematics

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

q ↦ u − 1 q u

is an ordinary or hyperbolic rotation according as

u = e a v , v ∈ I or u = e a p , p ∈ J .

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.

Reluctance 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 tt* = −1, then there is a p = i sinh(a) + r cosh(a) ∈ J such that t ∈ D_p, and an b ∈ R such that t = p exp(bp). Then if u = exp(bp), v = i cosh(a) + r sinh(a), 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.

Synonyms

Para-quaternions (Ivanov and Zamkovoy 2005, Mohaupt 2006) Manifolds with para-quaternionic structures are studied in differential geometry and string theory. In the para-quaternionic literature k is replaced with −k.

Exspherical system (Macfarlane 1900)

Split-quaternions (Rosenfeld 1988)

Antiquaternions (Rosenfeld 1988)

Pseudoquaternions (Yaglom 1968 Rosenfeld 1988)