Split-quaternion


×1ijk
11ijk
ii−1k−j
jj−k1−i
kkji1

In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. But unlike the quaternions, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. As an algebra over the real numbers, they are isomorphic to the algebra of 2 × 2 real matrices. For other names for split-quaternions see the [|Synonyms] section below.
The set forms a basis. The products of these elements are
and hence ijk = 1. It follows from the defining relations that the set is a group under split-quaternion multiplication; it is isomorphic to the dihedral group D4, the.
A split-quaternion
Due to the anti-commutative property of its basis vectors, the product of a split-quaternion with its conjugate is given by an isotropic quadratic form:
Given two split-quaternions p and q, one has, showing that N is a quadratic form admitting composition. This algebra is a composition algebra and N is its norm. Any such that is a null vector, and its presence means that split-quaternions form a "split composition algebra" – hence their name.
When the norm is non-zero, then q has a multiplicative inverse, namely q∗/N. The set
is the set of units. The set P of all split-quaternions forms a ring with group of units. The split-quaternions with form a non-compact topological group, shown below to be isomorphic to.
Historically split-quaternions preceded Cayley's matrix algebra; split-quaternions evoked the broader linear algebra.

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
represents q in the ring of matrices: the multiplication of split-quaternions behaves the same way as the matrix multiplication. For example, the determinant of this matrix is
The appearance of the minus sign distinguishes splitquaternions from the quaternions, which have a plus sign here. The matrices of determinant one form the special unitary group SU, which are the split-quaternions of norm one, and provide the hyperbolic motions of the Poincaré disk model of hyperbolic geometry.
Besides the complex matrix representation, another linear representation associates split-quaternions with 2 × 2 real matrices. This isomorphism can be made explicit as follows: Note first the product
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. One can make the matrix product above correspond to in the split-quaternion ring. Then for an arbitrary matrix there is the bijection
which is in fact a ring isomorphism. Furthermore, computing squares of components and gathering terms shows that, which is the determinant of the matrix. Consequently, there is a group isomorphism between the unit quasi-sphere of split-quaternions and and hence also with : 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 split-quaternions is equal to the product of the two separate norms. Thus split-quaternions 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
to be used when producing the doubled product in the real-split cases. As before, the doubled conjugate so that
If a and b are split-complex numbers and split-quaternion
then

Profile

The subalgebras of P may be seen by first noting the nature of the subspace Let
The parameters z and r 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 split-quaternions
These are the equilateral-hyperboloidal coordinates described by Alexander Macfarlane and Carmody.
Next, form three foundational sets in the vector-subspace of the ring:
Now it is easy to verify that
and that
These set equalities mean that when p ∈ J then the plane
is a subring of P that is isomorphic to the plane of split-complex numbers just as when v is in I then
is a planar subring of P that is isomorphic to the ordinary complex plane C.
Note that for every, so that and are nilpotents. The plane is a subring of P that is isomorphic to the dual numbers. Since every coquaternion must lie in a Dp, a Cv, or an N plane, these planes profile P. For example, the unit quasi-sphere
consists of the "unit circles" in the constituent planes of P: In Dp it is a unit hyperbola, in N the "unit circle" is a pair of parallel lines, while in Cv it is indeed a circle.These ellipse/circles found in each Cv 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 split-quaternion, then the scalar part of q is w.
Definition. For non-zero split-quaternions q and t we write when the scalar part of the product qt∗ is zero.
Proof: ∗ = q follows from ∗ = u∗t∗, which can be established using the anticommutative property of vector cross products.

Counter-sphere geometry

The quadratic form qq∗ is positive definite on the planes Cv and N. Consider the counter-sphere.
Take m = x + yi + zr where r = j cos + k sin. Fix θ and suppose
Since points on the counter-sphere must line on the conjugate of the unit hyperbola in some plane, m can be written, for some
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:
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
is an ordinary or hyperbolic rotation according as
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 split-quaternions for kinematic models may stem from the signature when spacetime is presumed to have signature or. Nevertheless, a transparently relativistic kinematics appears when a point of the counter-sphere is used to represent an inertial frame of reference. Indeed, if, then there is a such that, and a such that. Then if,, and, the set is a pan-orthogonal basis stemming from t, and the orthogonalities persist through applications of the ordinary or hyperbolic rotations.

Historical notes

The coquaternions were initially introduced in 1849 by James Cockle in the London-Edinburgh-Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 Bibliography of the Quaternion Society. Alexander Macfarlane called the structure of split-quaternion vectors an exspherical system when he was speaking at the International Congress of Mathematicians in Paris in 1900.
The unit sphere was considered in 1910 by Hans Beck. For example, the dihedral group appears on page 419. The split-quaternion structure has also been mentioned briefly in the Annals of Mathematics.

Synonyms