Structure of the Cayley-Dickson algebras

TL;DR

This work develops a graded Cayley-Dickson framework to rigorously classify associativity and non-associativity across the Cayley-Dickson ladder, introducing three associativity types and four non-associativity types. It constructs dual graded and binary pyramids, defines blades and cycles, and proves a Cycle Theorem that enumerates triads and their cycle structure across octonions and ultracomplex algebras, revealing subalgebras isomorphic to ${\mathbb O}$ and pseudo-octonions ${\mathbb P_k}$. A zero-divisor counting theory yields the formula $Z_m = \frac{1}{16}(N_m-1)(N_m-3)(N_m-7)$ with $N_m = 2^{m+3}-1$, and classifies divisors via silos, showing multiples of 84 divisors at higher levels. The paper also develops Split Cayley-Dickson algebras, showing a single split algebra per level with unitary generators $u_i$ and pure-trace properties, and computes zero-divisor counts for split cases (e.g., 12 for ${\widetilde{\mathbb O}}$, 180 for ${\widetilde{\mathbb U}_1}$), along with idempotents and nilpotents. Overall, the graded framework clarifies the non-associative landscape, ties to Malcev/Moufang identities, and connects subalgebra structures to potential physical applications, while posing open questions about the pseudo-octonion family.

Abstract

Viewing the Cayley-Dickson process as a graded construction provides a rigorous definition of associativity consisting of three types 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. This structure is distinct for the first four power-associative algebras and has the same pattern for at least the next three. The structure is identified with 3-cycles and modes that reduce sets of 84 zero divisors to just 7, in most cases, such as for the sedenions, and identifies the subalgebas of the power associative algebras that provide zero divisors thus defining the structure of Cayley-Dickson algebras. The zero divisor and octonion cardinality at all levels is derived. Split algebras are proved to be isomorphic for any power-associative algebra and share the same non-associative structure with different zero divisors. Due to the inclusion of new power-associative algebras into the hypercomplex algebras the terminology ultracomplex numbers is suggested.

Figures (2)

• Figure 1: Injections of Cayley Dickson algebras
• Figure 2: Special mode transformations

Theorems & Definitions (53)

• Definition
• Lemma 1
• proof
• Definition
• Definition
• Definition
• Definition
• Lemma 2
• proof
• Lemma 3