3-Parameter Generalized Quaternions (original) (raw)

Generalized Quaternions and Matrix Algebra

Fen ve mühendislik bilimleri dergisi, 2023

Bu çalışmada, Hamilton operatörlerini kullanarak genelleştirilmiş kuaterniyon cebiri ile gerçel (kompleks) matris cebirleri arasındaki bağlantıyı kurduk. Genelleştirilmiş kuaterniyonların gerçel ve kompleks temeline karşılık gelen gerçel ve kompleks matrisler elde ettik. Ayrıca, gerçel ve kompleks matrislerin temel özelliklerini araştırdık. Genelleştirilmiş kuaterniyonlara karşılık gelen Pauli matrislerini elde ettik. Daha sonra, bu matrisler tarafından üretilen cebirin, genelleştirilmiş 3 uzayı tarafından üretilen Clifford cebiri (3) ile izomorf olduğunu gösterdik. Son olarak, genelleştirilmiş birim kuaterniyonlara karşılık gelen simplektik matrisler grubu, genelleştirilmiş birim matrisler grubu ve genelleştirilmiş ortogonal matrisler grubu arasındaki ilişkileri inceledik.

A Generalization of Quaternions and Their Applications

Symmetry, 2022

There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, and the other six are generalizations of the split-quaternion concept first introduced by Cockle. We show that the 4×4 matrix representations of both the left- and right-chirality versions of the generalized split-quaternions are algebraically isomorphic and can be related to each other by 4×4 permutation matrices of the C2×C2 group. As examples of applications of the generalized quaternion concept, we first show that the left- and right-chirality quaternions can be used to describe Lorentz transformations with a constant velocity in an arbitrary spatial direction. Then, it is shown how each of the generalized split-quaternion algebras can be used to solve the problem of quantum-mechanical ...

Understanding Quaternions from Modern Algebra and Theoretical Physics

2021

Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius’ theorem, which says that “ the only finite-dimensional real division algebra are the real field R, the complex field C and the algebra H of quaternions” was derived. They appear also through Hamilton formulation of mechanics, as elements of rotation groups in the symplectic vector spaces. Quaternions were used in the solution of 4-dimensional Dirac equation in QED, and also in solutions of Yang-Mills equation in QCD as elements of noncommutative geometry. We present how quaternions are formulated in Clifford Algebra, how it is used in explaining rotation group in symplectic vector space and parallel transformation in holonomic dynamics. When a dynamical system *E-mail address: furui@umb.teikyo-u.ac.jp.

Generalized quaternions and spacetime symmetries

Journal of Mathematical Physics, 1982

The construction of a class of associative composition algebras qn on R 4 generalizing the wellknown quaternions Q provides an explicit representation of the universal enveloping algebra of the real three-dimensional Lie algebras having tracefree adjoint representations (class A Bianchi type Lie algebras). The identity components of the four-dimensional Lie groups GL(qn,l) Cqn (general linear group in one generalized quaternion dimension) which are generated by the Lie algebra of this class of quaternion algebras are diffeomorphic to the manifolds of spacetime homogeneous and spatially homogeneous spacetimes having simply transitive homogeneity isometry groups with tracefree Lie algebra adjoint representations. In almost all cases the complete group ofisometries of such a spacetime is isomorphic to a subgroup of the group ofleft and right translations and automorphisms of the appropriate generalized quaternion algebra. Similar results hold for the single class B Lie algebra of Bianchi type V, characterized by its "pure trace" adjoint representation.

Some results on generalized quaternions algebra with generalized Fibonacci quaternions

Annals of the Alexandru Ioan Cuza University - Mathematics

Recently, the most general form of the quaternion algebra on 3 parameters (3PGQ) has been introduced, which prompted us to look for some properties associated with this algebra, which will be called k λ 1 ,λ 2 ,λ 3 in this article. This paper consists of two main parts: the first part focuses on derivations, while the second part deals with the Fibonacci sequence, and the properties of both parts are related to the algebra k λ 1 ,λ 2 ,λ 3. First we determine the derivations of the algebra k λ 1 ,λ 2 ,λ 3 and get the algebra Der(k λ 1 ,λ 2 ,λ 3) of derivations of kλ 1 ,λ 2 ,λ 3. Then we were able to obtain generalized derivations that have been studied by mathematicians in the context of algebras of certain normed spaces, and semi-prime and prime rings. In the second part, we introduce some properties of the Fibonacci quaternions in the generalized quaternion algebra k λ 1 ,λ 2 ,λ 3 , which allow us to assume that the algebra k λ 1 ,λ 2 ,λ 3 is not always a division algebra.

Third-Order Jacobsthal Generalized Quaternions

Journal of Geometry and Symmetry in Physics

In this paper, the third-order Jacobsthal generalized quaternions are introduced. We use the well-known identities related to the thirdorder Jacobsthal and third-order Jacobsthal-Lucas numbers to obtain the relations regarding these quaternions. Furthermore, the third-order Jacobsthal generalized quaternions are classified by considering the special cases of quaternionic units. We derive the relations between third-order Jacobsthal and third-order Jacobsthal-Lucas generalized quaternions.

Homothetic exponential motions with generalized quaternions

Pure Mathematical Sciences, 2014

In this paper, we give the real matrix representations of generalized quaternions, and then, the concept of homothetic exponential motions is discussed and their velocities, accelerations obtained, Also, the paper gives some formula and facts about exponential motions which are not generally known for generalized quaternions.

A New Polar Representation for Split and Dual Split Quaternions

Advances in Applied Clifford Algebras, 2017

We present a new different polar representation of split and dual split quaternions inspired by the Cayley-Dickson representation. In this new polar form representation, a split quaternion is represented by a pair of complex numbers, and a dual split quaternion is represented by a pair of dual complex numbers as in the Cayley-Dickson form. Here, in a split quaternion these two complex numbers are a complex modulus and a complex argument while in a dual split quaternion two dual complex numbers are a dual complex modulus and a dual complex argument. The modulus and argument are calculated from an arbitrary split quaternion in Cayley-Dickson form. Also, the dual modulus and dual argument are calculated from an arbitrary dual split quaternion in Cayley-Dickson form. By the help of polar representation for a dual split quaternion, we show that a Lorentzian screw operator can be written as product of two Lorentzian screw operators. One of these operators is in the two-dimensional space produced by 1 and i vectors. The other is in the three-dimensional space generated by 1, j and k vectors. Thus, an operator in a four-dimensional space is expressed by means of two operators in two and three-dimensional spaces. Here, vector 1 is in the intersection of these spaces.

Vector Representation of Quaternions Solutions of Quaternionic Quadratic Equations

Journal of Pure and Applied Mathematics: Advances and Applications, 2015

The fundamental theorem of algebra over the quaternion skew field has gained little attention. It is of less importance than that over the complex number field, though some problems may reduce to it. In this work, we use the vector representation of quaternions in order to obtain simplified forms of the solutions of the quaternionic equation 0 2