Involutions in split semi‐quaternions

Abstract

A map is an involution (resp, anti-involution) if it is a self-inverse homomorphism (resp, antihomomorphism) of a field algebra. The main purpose of this paper is to show how split semi-quaternions can be used to express half-turn planar rotations in 3-dimensional Euclidean space R 3 and how they can be used to express hyperbolic-isoclinic rotations in 4-dimensional semi-Euclidean space R 3;1 . We present an involution and an anti-involution map using split semi-quaternions and give their geometric interpretations as half-turn planar rotations in R 3 . Also, we give the geometric interpretation of nonpure unit split semi-quaternions, which are in the form p = coshθ + sinhθi + 0j + 0k = coshθ + sinhθi, as hyperbolic-isoclinic rotations in R 3;1 .

Generalized tubes in pseudo‐Galilean 3‐space: Split semi‐quaternionic representations and an application to magnetic flux tubes

“…Now, we shall introduce the concept of a split semi‐quaternion and present some properties of such quaternions. For more information, please see previous studies 41–44 …”

On quaternion‐Gaussian Lucas numbers

“…Quaternion‐Gaussian numbers are used for investigating graphical models 10 . Therefore, these numbers can be used in other fields such as previous studies 11,12 …”

“…10 Therefore, these numbers can be used in other fields such as previous studies. 11,12 Inspired by these studies, we have described a new type of quaternion sequence in this paper and called the elements of this sequence as quaternion-Gaussian Lucas numbers GLQ n .…”