Cayley-Dicksonia Revisited

TL;DR

This paper revisits the Cayley-Dickson process, presenting a generalized form that allows for nonstandard multiplication and broadens the algebraic framework of hypercomplex systems.

Contribution

It introduces a generalized Cayley-Dickson process based on units not necessarily tied to orthogonal vector spaces, expanding the algebraic formulations of hypercomplex numbers.

Findings

Generalized Cayley-Dickson process developed

Matrices with nonstandard multiplication analyzed

Broader algebraic structures for hypercomplex systems proposed

Abstract

In the theory of the hypercomplex, the laws governing the algebra are based on units that are naturally associated with an orthogonal vector space, a requirement that is far from mandatory in many algebraic formulations arising in the context of the reals or the complex numbers.In this article the complementing view is held, in that the laws of hypercomplex algebra are recast in terms of quite generally posited units. Proceeding in this manner, a generalized form of the Cayley-Dickson process is examined. The representations given are regular bimodular; the resulting matrices are standard except they are allowed nonstandard multiplication for noncommutative matrix elements.