Split-quaternion

In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.

After introduction in the 20th century of coordinate-free definitions of rings and algebras, it has been proved that the algebra of split-quaternions is isomorphic to the ring of the 2×2 real matrices. So the study of split-quaternions can be reduced to the study of real matrices, and this may explain why there are few mentions of split-quaternions in the mathematical literature of the 20th and 21st centuries.

The split-quaternions are the linear combinations (with real coefficients) of four basis elements 1, i, j, k that satisfy the following product rules:

Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real associative algebra. But like the matrices and unlike the quaternions, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. (For example, 1/2(1 + j) is an idempotent zero-divisor, and i − j is nilpotent.) As an algebra over the real numbers, the algebra of split-quaternions is isomorphic to the algebra of 2×2 real matrices by the above defined isomorphism.

This isomorphism allows identifying each split-quaternion with a 2×2 matrix. So every property of split-quaternions corresponds to a similar property of matrices, which is often named differently.

The conjugate of a split-quaternion q = w + xi + yj + zk, is q = wxi − yj − zk. In term of matrices, the conjugate is the cofactor matrix obtained by exchanging the diagonal entries and changing of sign the two other entries.

The product of a split-quaternion with its conjugate is the isotropic quadratic form:

which is called the norm of the split-quaternion or the determinant of the associated matrix.

The real part of a split-quaternion q = w + xi + yj + zk is w = (q + q)/2. It equals the trace of associated matrix.

The norm of a product two split-quaternions is the product of their norms. Equivalently, the determinant of a product of matrices is the product of their determinants.

This means that split-quaternions and 2×2 matrices form a composition algebra. As there are nonzero split-quaternions having a zero norm, split-quaternions form a "split composition algebra" – hence their name.

A split-quaternion with a nonzero norm has a multiplicative inverse, namely q/N(q). In terms of matrix, this is Cramer rule that asserts that a matrix is invertible if and only its determinant is nonzero, and, in this case, the inverse of the matrix is the quotient of the cofactor matrix by the determinant.

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, which must not be confused with the basic split quaternion i (upright roman).

The image of this homomorphism is the matrix ring formed by the matrices of the form

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. Let S be the matrix

Then, applied to the representation of split-quaternions as 2×2 real matrices, the above algebra homomorphism is the matrix similarity.

It follows almost immediately that for a split quaternion represented as a complex matrix, the conjugate is the matrix of the cofactors, and the norm is the determinant.

With the representation of split quaternions as complex matrices. the matrices of quaternions of norm 1 are exactly the elements of the special unitary group SU(1,1). This is used for in hyperbolic geometry for describing hyperbolic motions of the Poincaré disk model.[1]

Kevin McCrimmon [2] 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

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

Three cases have to be considered, which are detailed in the next subsections.

This is a parametrization of all split-quaternions whose nonreal part is nilpotent.

This is a parametrization of all split-quaternions whose nonreal part has a positive norm.

This is a parametrization of all split-quaternions whose nonreal part has a negative norm.

As seen above, the purely nonreal split-quaternions of norm –1, 1 and 0 form respectively a hyperboloid of one sheet, a hyporboloid of two sheets and a circular cone in the space of non real quaternions.

These surfaces are pairwise asymptote and do not intersect. 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. All these hyperboloids are asymptote to the above cone, and none of these surfaces intersect any other. 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)[3] in 1849 by James Cockle in the London–Edinburgh–Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 Bibliography[4] 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.[5]

The unit sphere was considered in 1910 by Hans Beck.[6] For example, the dihedral group appears on page 419. The split-quaternion structure has also been mentioned briefly in the Annals of Mathematics.[7][8]