Lie Algebra of Unit Tangent Bundle

Abstract

In this paper, semi-quaternions are studied with their basic properties. Unit tangent bundle of \({{\mathbb {R}}^2}\) is also obtained by using unit semi-quaternions and it is shown that the set \({{T {\mathbb {R}}^2}}\) of all unit semi-quaternions based on the group operation of semi-quaternion multiplication is a Lie group. Furthermore, the vector space matrix of angular velocity vectors forming the Lie algebra \({{T_{1} {\mathbb {R}}^2}}\) of the group \({{T {\mathbb {R}}^2}}\) is obtained. Finally, it is shown that the rigid body displacements obtained by using semi-quaternions correspond to planar displacements in \({{\mathbb {R}^3}}\).