Split-quaternion - Matrix Representations

Matrix Representations

Let

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

,

with (complex conjugates of u and v), represents q in the ring of matrices in the sense that 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, where there is a plus in H, distinguishes coquaternions from quaternions. The use of the split-quaternions of modulus one (q q* = 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

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

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 coquaternions and SL2(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 (1985) for the hyperbolic motion group representation with 2 × 2 real matrices.

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