Automorphisms of Sedenions

Abstract

This paper extends the 7-vertex Fano plane to a 15-vertex Fano volume, which is related to sedenions. The Fano plane visualises the octonions and their structure as seven quaternions and is derived from a calibration in differential geometry. Clifford algebra allows the Spin and Pin groups to fully analyse calibrations, which categorises ideals and creates a new calibration that derives the Fano volume and provides a visualisation of the complete subalgebra structure of sedenions. This new 15-dimensional calibration, along with the existing associative one containing 35 quaternions, enables an explicit derivation of the automorphisms of sedenions extending Schafer's and Brown's results and addressing a discrepancy in the literature. Also, Gresnigt has suggested extending Furey's work on fermion families using octonions to three octonion subalgebras of sedenions. The Fano volume demonstrates visually the process to avoid the power-associative subalgebras of sedenions. It is conjectured that Fano hyper-volumes will uncover the complete subalgebra structure and automorphisms of all Cayley-Dickson algebras.