On the metric hypercomplex group alternative-elastic algebras for n mod 8 = 0

TL;DR

This paper defines hypercomplex orthogonal algebras and shows their relation to metric hypercomplex group alternative-elastic algebras for dimensions divisible by 8, including explicit constructions and a Delphi implementation for sedenions.

Contribution

It introduces a new framework for hypercomplex orthogonal algebras, linking them to metric hypercomplex group alternative-elastic algebras and providing explicit generator forms and implementation.

Findings

Hypercomplex orthogonal algebra is a metric hypercomplex group alternative-elastic algebra for n mod 8=0.

Generated by a symmetric (0,2)-spinor and orthogonal transformations.

Explicit generator form and Delphi implementation for n=16 sedenions.

Abstract

In this article the hypercomplex orthogonal (homogenous) algebra definition is made. It is shown that 1. the hypercomplex orthogonal algebra is the metric hypercomplex group alternative-elastic algebra for n mod 8 = 0 (the non-alternative and non-normalized, but the weakly alternative and weakly normalized for n>8; the alternative and normalized for the oktonion algebra); 2. the hypercomplex orthogonal algebra is generated by a symmetric (0,2)-spinor; 3. the hypercomplex orthogonal homogeneous algebra is generated by the identity algebra, generating algebra and orthogonal transformations; 4. the metric hypercomplex Cayley-Dickson algebra is the hypercomplex special orthogonal homogeneous algebra for n=2^k,n>=8. The hypercomplex Cayley-Dickson algebra generator in an explicit form is calculated. Technical realization (Delphi) of the canonical sedenion algebra for n=16 is given according to the point 2-3.