Automorphisms of Sedenions

Abstract

This paper extends the octonion calibration in Clifford algebra to two related sedenion calibrations. Associative calibrations map quaternion rings in Clifford algebra to those in Cayley-Dickson algebras, with octonions consisting of seven rings and sedenions having 35 rings. A new non-associative calibration is found that is related to the sedenion associative calibration but is invertible, providing a classification of the ideals of the even sub-algebra of Clifford algebra. This leads to subalgebras of certain Clifford algebras and a thorough analysis of the possible automorphisms of sedenions is applied using the Spin and Pin groups. While the calibrations and automorphsms of octonions have minimal divergence, it is found sedenions introduce new algebras and this work clarifies the discrepancy in the results of Schafer and Brown on the automorphisms of sedenions. The non-associative calibrations are found to be an infinite series that matches the finite geometry PG(N,2) series, discovered by Gino Fano, but since this series can be derived from simplices the name Fano hyper-volumes is suggested. The visual representation of sedenions as the 3-D Fano volume, representing the 15 Fano planes of sedenions, allows the power-associative rings to be visually distinguished from the octonion non-associative rings. These are more than loops because the power-associative rings of sedenions contain zero divisors. The visual representation of sedenions may be important for particle physics and Clifford algebra provides analysis tools for the PG(N,2) series that uncovers the hidden structure of Cayley-Dickson algebras within an associative algebra.