Split-quaternion

The split-quaternions are the linear combinations (with real coefficients) of four basis elements 1, i, j, k that satisfy the following product rules: By associativity, these relations imply and also ijk = 1.

So, the split-quaternions form a real vector space of dimension four with {1, i, j, k} as a basis.

They form also a noncommutative ring, by extending the above product rules by distributivity to all split-quaternions.

Let consider the square matrices They satisfy the same multiplication table as the corresponding split-quaternions.

(respectively) induces an algebra isomorphism from the split-quaternions to the two-by-two real matrices.

In fact, if one considers a square whose vertices are the points whose coordinates are 0 or 1, the matrix

Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real associative algebra.

The real part of a split-quaternion q = w + xi + yj + zk is w = (q∗ + q)/2.

This property means that split-quaternions form a composition algebra.

Geometrically, the split-quaternions can be compared to Hamilton's quaternions as pencils of planes.

For split-quaternions there are hyperboloids of hyperbolic and imaginary units that generate split-complex or ordinary complex planes, as described below in § Stratification.

There is a representation of the split-quaternions as a unital associative subalgebra of the 2×2 matrices with complex entries.

This representation can be defined by the algebra homomorphism that maps a split-quaternion w + xi + yj + zk to the matrix Here, i (italic) is the imaginary unit, not to be confused with the split quaternion basis element i (upright roman).

This homomorphism maps respectively the split-quaternions i, j, k on the matrices The proof that this representation is an algebra homomorphism is straightforward but requires some boring computations, which can be avoided by starting from the expression of split-quaternions as 2×2 real matrices, and using matrix similarity.

the matrices of quaternions of norm 1 are exactly the elements of the special unitary group SU(1,1).

[1] Split-quaternions may be generated by modified Cayley–Dickson construction[2] similar to the method of L. E. Dickson and Adrian Albert.

In this section, the real subalgebras generated by a single split-quaternion are studied and classified.

Therefore, there are real numbers n, t, u such that 0 ≤ t < 2π and This is a parametrization of all split-quaternions whose nonreal part has a positive norm.

This is also a parameterization of the corresponding subalgebras by the pairs of opposite points of a hyperboloid of two sheets: the split-quaternions of the form

The algebra generated by a split-quaternion with a nonreal part of positive norm is isomorphic to

Therefore, there are real numbers n, t, u such that 0 ≤ t < 2π and This is a parametrization of all split-quaternions whose nonreal part has a negative norm.

This is also a parameterization of the corresponding subalgebras by the pairs of opposite points of a hyperboloid of one sheet: the split-quaternions of the form

The algebra generated by a split-quaternion with a nonreal part of negative norm is isomorphic to

Their complement consist of six connected regions: This stratification can be refined by considering split-quaternions of a fixed norm: for every real number n ≠ 0 the purely nonreal split-quaternions of norm n form an hyperboloid.

As the set of the purely nonreal split-quaternions is the disjoint union of these surfaces, this provides the desired stratification.

The coquaternions were initially introduced (under that name)[4] in 1849 by James Cockle in the London–Edinburgh–Dublin Philosophical Magazine.

The introductory papers by Cockle were recalled in the 1904 Bibliography[5] 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.

[6] Macfarlane considered the "hyperboloidal counterpart to spherical analysis" in a 1910 article "Unification and Development of the Principles of the Algebra of Space" in the Bulletin of the Quaternion Society.

[8] The split-quaternion structure has also been mentioned briefly in the Annals of Mathematics.