Generalized quaternions and spacetime symmetries

Trends in Mathematics, 2013

The present paper is aimed at proving necessary and sufficient conditions on the quaternionic-valued coefficients of a first-order linear operator to be associated to the generalized Cauchy-Riemann operator in quarternionic analysis and explicitly we give the description of all its nontrivial first-order symmetries.

1994

A global model of the q deformation for the quasiorthogonal Lie algebras generating the groups of motions of the four-dimensional affine Cayley-Klein (CK) geometries is obtained starting from the three-dimensional deformations. It is shown how the main algebraic classical properties of the CK systems can be implemented in the quantum case. Quantum deformed versions either of the space-time or space symmetry algebras [Poincare (3+ l), Galilei (3+ l), 4-D Euclidean as well as others] appear in this context as particular cases. For some of these classical algebras several q deformations are directly obtained.

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 ...

Annali di Matematica Pura ed Applicata, 1982

Texts in Applied Mathematics, 2011

One of the main goals of these notes is to explain how rotations in R n are induced by the action of a certain group, Spin(n), on R n , in a way that generalizes the action of the unit complex numbers, U(1), on R 2 , and the action of the unit quaternions, SU(2), on R 3

arXiv:2310.02114, 2023

We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel with respect to the Cartan-Schouten canonical connection) on perfect Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T∗G of a Lie group G, are always endowed with their Lie group structures induced by the right trivialization. We show that TG and T∗G are isomorphic if G possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to T∗SU(2), the set of unit dual split quaternions is isomorphic to T∗SL(2, R). The group SE(3) of special rigid displacements of the Euclidean 3-space is isomorphic to T∗SO(3). The group SE(2, 1) of special rigid displacements of the Minkowski 3-space is isomorphic to T∗SO(2,1). Some results on SE(3) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to SE(2, 1) and to T∗G, for any simple Lie group G.

2021

In this paper, we use four-dimensional quaternionic algebra to describing space-time geometry in curvature form. The transformation relations of a quaternionic variable are established with the help of basis transformations of quaternion algebra. We deduced the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields. The quaternionic metric tensor and geodesic equation are also discussed to describing the quaternionic line element in curved space-time. Moreover, we discussed an expression for the Riemannian Christoffel curvature tensor in terms of the quaternionic metric tensor. We have deduced the quaternionic Einstein’s field-like equation which shows an equivalence between quaternionic matter and geometry.

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: [email protected].

1999

CP 6065, 13081-970 Campinas (SP) Brasil deleo/

2005

In this paper, we continue the study of the Killing symmetries of a N-dimensional generalized Minkowski space, i.e. a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical coordinates. We discuss here the finite structure of the space-time rotations in such spaces, by confining ourselves (without loss of generality) to the four-dimensional case. In particular, the results obtained are specialized to the case of a ''deformed'' Minkowski space $% \widetilde{M_{4}}$ (i.e. a pseudoeuclidean space with metric coefficients depending on energy), for which we derive the explicit general form of the finite rotations and boosts in different parametric bases.