Split quaternions and semi-Euclidean projective spaces

In this study, we give one-to-one correspondence between the elements of the unit split three-sphere Sð3; 2Þ with the complex hyperbolic special unitary matrices SUð2; 1Þ. Thus, we express spherical concepts such as meridians of longitude and parallels of latitude on SUð2; 1Þ by using the method given in Toth [Toth G. Glimpses of algebra and geometry. Springer-Verlag; 1998] for S3. The relation among the special orthogonal group SOðR3Þ, the quotient group of unit quaternions S3=f 1g and the projective space RP3 given as SOðR3Þ ffi S3=f 1g ¼ RP3 is known as the Euclidean projective spaces [Toth G. Glimpses of algebra and geometry. Springer- Verlag; 1998]. This relation was generalized to the semi-Euclidean projective space and then, the expression SOð3; 1Þ ffi Sð3; 2Þ=f 1g ¼ RP32 was acquired. Thus, it was found that Hopf fibriation map of Sð2; 1Þ can be used for Twistors (in not-null state) in quantum mechanics applications. In addition, the octonions and the split-octonions can be obtained from the Cayley-Dickson construction by defining a multiplication on pairs of quaternions or split quaternions. The automorphism group of the octonions is an exceptional Lie group. The split-octonions are used in the description of physical law. For example, the Dirac equation in physics (the equation of motion of a free spin 1/2 particle, like e.g. an electron or a proton) can be represented by a native split-octonion arithmetic.