in this paper, lie group structure and lie algebra structure of unit complex 3-sphere     are studied. in order to do this, adjoint representations of unit biquaternions (complexified quaternions) are obtained. also, a correspondence between the elements of     and the special complex unitary matrices    (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers    .  the relat...

The roots of -1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are studied and it is shown that there is an infinite number of non-trivial complexified quaternion roots (and two degenerate solutions which are the complex imaginary operator and the set of unit pure real quaternions). The non-trivial roots are shown t...

A. F. Horadam defined the complex Fibonacci numbers and quaternions in middle of 20th century. Half a century later, S. Hal{\i}c{\i} introduced by inspiring from these definitions discussed some properties them. Recently, elliptic biquaternions, which are generalized form real quaternions, have been presented. In this study, we introduce set biquaternions that includes as special case investiga...

We develop a relativistic free wave equation on the complexified quaternions, linear in the derivatives. Even if the wave functions are only one-component, we show that four independent solutions, corresponding to those of the Dirac equation, exist. A partial set of translations between complex and complexified quaternionic quantum mechanics may be defined. a) e-mail address: [email protected]

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for suitably defined self-adjoint complex 4-velocity, pure can be phrased in terms the quaternion square root relative 4-velocity connecting two inertial frames. Straightforward computations then lead to quite explicit relatively simple algebraic formulae ...

We reformulate Special Relativity by a quaternionic algebra on reals. Using real linear quaternions, we show that previous difficulties, concerning the appropriate transformations on the 3 + 1 space-time, may be overcome. This implies that a complexified quaternionic version of Special Relativity is a choice and not a necessity. a) e-mail: [email protected]

The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and ”tensor” formulation of Q-units with their possible representations are discussed and groups of Q-units transformations leaving Q-multiplication rule form-invariant are determined. A series of mathematical and physical applications is offered, among...