An Introduction to Newly Defined Quaternions: Split-like Quaternions

Abstract

In this paper, we define split-like quaternions such that p = p0 + p1i + p2j + p3k, where i2 =j2 = -1, k2 = 1 and p0, p1, p2, p3 is an element of I. It is the purpose of this paper to describe a new non-commutative and non-associative quaternion algebra which is isomorphic to the four-dimensional Lorentz space I41. In addition, we define the inner product, characteristic of split-like quaternions, cross product and we discuss their properties. Furthermore, we give the matrix representations of split-like quaternions. Also, Cauchy-Schwarz inequality and triangle inequality provided by the inner product are demonstrated. Finally, we present the polar forms of split-like quaternions by giving the angle between two split-like quaternions and give De Moivre's formulas for split-like quaternions.