Symmetries and Associated Pairs in Quaternionic Analysis

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.

Mathematische Nachrichten, 2015

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field H. It relies heavily on results on functions defined on domains in R 4 or R 3 with values in H. This theory is centred around the concept of ψ-hyperholomorphic functions related to a so-called structural set ψ of H 4 or H 3 respectively. The main goal of this paper is to develop the nucleus of the (ϕ, ψ)-hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy-Riemann operators associated to a pair ϕ, ψ of structural sets of H 4. Following a matrix approach, a generalized Borel-Pompeiu formula and the corresponding Plemelj-Sokhotzki formulae are established.

2000

Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential equations with constant coefficients. We overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients. The resolution of the complex linear Schrodinger equation, in presence of quaternionic potentials, represents an interesting application of the mathematical material discussed in this paper.

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 ...

We draw the conclusions from the earlier presented quaternionic generalization of Cauchy-Riemann's equations. The general expressions for constituents of -holomorphic functions as well as the relations between them are deduced. The symmetry properties of constituents of -holomorphic functions and their derivatives of all orders are proved. For full derivatives it is a consequence of uniting the left and right derivatives within the framework of the developed theory. Some -holomorphic generalizations of ℂ − holomorphic functions are discussed in detail to demonstrate particularities of constructing H-holomorphic functions. The power functions are considered in detail.

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].

2003

In a recent paper [J.Math.Phys. vol42, 2236-2265 (2001)], we discussed differential operators within a quaternionic formulation of quantum mechanics. In particular, we proposed a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients. In this paper, we extend our discussion to real linear quaternionic differential equations. The method of resolution is based on the Jordan canonical form of quaternionic matrices associoated to real linear differential operators.

Computers & Mathematics With Applications, 2004

Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential equations with constant coefficients. We overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients. The resolution of the complex linear Schrödinger equation, in presence of quaternionic potentials, represents an interesting application of the mathematical material discussed in this paper.

This is a compilation of quaternionic number systems, quaternionic function theory, quaternionic Hilbert spaces and Gelfand triples. The difference between quaternionic differential calculus and Maxwell based differential calculus is explained.

Annali di Matematica Pura ed Applicata, 1982