Discrete octonionic analysis: a unified approach to the split-octonionic and classical settings
Rolf Sören Kraußhar, Anastasiia Legatiuk, Dmitrii Legatiuk
TL;DR
The paper develops a discrete octonionic analysis framework that unifies classical and split-octonionic settings across multiple discretisations. It introduces abstract associators and a Stokes operator to create a general, algebra-agnostic toolkit that simultaneously covers $\mathbb{O}$ and $\mathbb{O}'$ and both forward/backward Cauchy–Riemann and Weyl-calculus approaches. The main contributions are the abstract Stokes and boundary operators, and the corresponding discrete Stokes, Borel–Pompeiu, and Cauchy formulae for whole space and half-spaces, with explicit forms involving $E_h^{+}$ and index-set distinctions $I_k$ vs $J_k$. The work provides a unified, practical framework that can be applied to eight-dimensional problems (e.g., dyonic plasma dynamics) and offers a consistent discretisation toolkit bridging continuous and discrete octonionic analysis in both algebraic settings.
Abstract
Various problems of mathematical physics consider octonions and split-octonions as a mathematical structure, which underpins the eight-dimensional nature of these problems. Therefore, it is not surprising that octonionic analysis has become an area of active research in recent years. One of the main goals of octonionic analysis is to develop tools of an octonionic operator calculus for solving boundary value problems of mathematical physics that benefit from the use of the octonionic structure. However, when we want to apply the operator calculus in practice, it becomes evident that adequate discrete counterparts of continuous constructions need to be defined. In previous works, we have proposed several approaches to discretise the classical continuous octonionic analysis. However, the split-octonionic case, which is particularly important for practical applications concretely investigated in the last years, has not been considered until now. Therefore, one of the goals of this paper is to explain how to particularly address the discrete split-octonionic setting. Additionally, we propose a general umbrella to cover all different discrete octonionic settings in one unified approach that also encompasses the different eight-dimensional algebraic structures.