Cayley-Dickson Algebras and Finite Geometry

TL;DR

This paper explores the deep connections between Cayley-Dickson algebras and finite projective geometries, revealing how algebraic properties encode geometric configurations and uncovering a nested pattern of combinatorial structures.

Contribution

It establishes a novel link between Cayley-Dickson algebras and finite geometries through Veldkamp spaces and configurations, identifying specific geometric structures for algebras up to dimension 64.

Findings

Identification of projective space PG(N-1,2) with Cayley-Dickson units

Description of configurations for octonions, sedenions, and higher algebras

Observation of nesting patterns and conjecture on Grassmannian isomorphism

Abstract

Given a $2^N$-dimensional Cayley-Dickson algebra, where $3 \leq N \leq 6$, we first observe that the multiplication table of its imaginary units $e_a$, $1 \leq a \leq 2^N -1$, is encoded in the properties of the projective space PG$(N-1,2)$ if one regards these imaginary units as points and distinguished triads of them $\{e_a, e_b, e_c\}$, $1 \leq a < b <c \leq 2^N -1$ and $e_ae_b = \pm e_c$, as lines. This projective space is seen to feature two distinct kinds of lines according as $a+b = c$ or $a+b \neq c$. Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG$(N-1,2)$, the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a binomial $\left({N+1 \choose 2}_{N-1}, {N+1 \choose 3}_{3}\right)$-configuration ${\cal C}_N$; in particular, ${\cal C}_3$ (octonions) is isomorphic to the Pasch $(6_2,4_3)$-configuration, ${\cal C}_4$ (sedenions) is the famous Desargues $(10_3)$-configuration, ${\cal C}_5$ (32-nions) coincides with the Cayley-Salmon $(15_4,20_3)$-configuration found in the well-known Pascal mystic hexagram and ${\cal C}_6$ (64-nions) is identical with a particular $(21_5,35_3)$-configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. We also draw attention to a remarkable nesting pattern formed by these configurations, where ${\cal C}_{N-1}$ occurs as a geometric hyperplane of ${\cal C}_N$. Finally, a brief examination of the structure of generic ${\cal C}_N$ leads to a conjecture that ${\cal C}_N$ is isomorphic to a combinatorial Grassmannian of type $G_2(N+1)$.