On algebras of three-dimensional quaternionic harmonic fields

TL;DR

This paper investigates the structure of quaternionic harmonic fields on 3D Riemannian manifolds, focusing on the existence of commutative algebras within them and exploring potential applications to impedance tomography.

Contribution

It characterizes conditions under which quaternionic harmonic fields form commutative algebras and discusses their possible use in impedance tomography.

Findings

Identification of conditions for algebraic structures within harmonic fields

Analysis of quaternionic harmonic fields on 3D Riemannian manifolds

Potential application insights for impedance tomography

Abstract

A quaternionic field is a pair $p=\{\alpha,u\}$ of function $\alpha$ and vector field $u$ given on a 3d Riemannian maifold $\Omega$ with the boundary. The field is said to be harmonic if $\nabla \alpha={\rm rot\,}u$\, in $\Omega$. The linear space of harmonic fields is not an algebra w.r.t. quaternion multiplication. However, it may contain the commutative algebras, what is the subject of the paper. Possible application of these algebras to the impedance tomography problem is touched on.