Structure of the Cayley-Dickson algebras

TL;DR

This paper explores the structure of Cayley-Dickson algebras using a graded approach, revealing new insights into associativity, zero divisors, and subalgebra embeddings, especially focusing on power-associative and ultracomplex numbers.

Contribution

It introduces a graded framework for Cayley-Dickson algebras, clarifies associativity classes, and systematically analyzes zero divisors and subalgebra structures, including split algebras.

Findings

Power-associativity introduces zero divisors in a structured way.

Zero divisors occur in multiples of 84, reducible to factors of seven.

Embedded subalgebras relate to ultracomplex numbers in multiples of seven.

Abstract

Viewing the Cayley-Dickson process as a graded construction provides a rigorous definition of associativity consisting of three classes and the non-associative parts dividing into four types. These simplify the Moufang loop identities and Mal'cev's identity, which identifies the non-associative Lie algebra structure. Analysing the non-associativity structure uncovers 3-cycles that distinguish between the Moufang identities and are used to identify three power-associative subalgebras of sedenions and higher level Cayley-Dickson algebras. Power-associativity introduces zero divisors into Cayley-Dickson algebras in a systematic way and it is convenient to replace the terminology hypercomplex numbers with {\it ultracomplex numbers} for the power-associative algebras. The non-associative types show that zero divisors in these algebras occur in multiples of 84 and cycles and modes are uncovered that reduce these down to factors of seven primary zero divisor pairs. It is shown that this is due to the power-associative subalgebras being embedded into ultracomplex numbers in multiples of seven. The graded notation allows the eight octonion and seven power-associative subalgebras of sedenions to be uniquely derived, up to representation. The zero divisors for split sedenion algebras are analysed and mappings between three of these are provided. These split algebras are shown to involve the same power-associative subalgebras as sedenions.