3-parameter generalized quaternions

TL;DR

This paper introduces the most general form of 3-parameter generalized quaternions, exploring their algebraic properties, matrix representations, and applications in Lie groups and algebras.

Contribution

It defines 3-parameter generalized quaternions (3PGQs) and studies their properties, matrix forms, and applications in Lie theory, extending classical quaternion concepts.

Findings

Defined 3PGQs and their algebraic operations

Derived matrix, polar, De Moivre's, and Euler's representations

Explored Lie group and Lie algebra structures associated with 3PGQs

Abstract

In this article, we give the most genaral form of the quaternions algebra depending on 3-parameters. We define 3-parameter generalized quaternions (3PGQs) and study on various properties and applications. Firstly we present the definiton, the multiplication table another properties of 3PGQs such as addition-substraction, multiplication and multiplication by scalar operations, unit and inverse elements, conjugate and norm. We give matrix representation and Hamilton operators for 3PGQs. We get polar represenation, De Moivre's and Euler's formulas with the matrix representations for 3PGQs. Besides, we give relations among the powers of the matrices associated with 3PGQs. Finally, Lie group and Lie algebra are studied and their matrix representations are shown. Also the Lie multiplication and the killing bilinear form are given.