Octonionic matrices larger than 3x3 do not make a Jordan algebra

In jordan: A Suite of Routines for Working with Jordan Algebras

knitr::opts_chunk$set(echo = TRUE)

We can set up an ersatz Jordan algebra as follows:

library(jordan)
`%o%` <- function(x,y){(x%*%y + y%*%x)/2}
ji <- function(x,y){   # ji == Jordan Identity; if satisfied exactly, return 0
   max(Mod((x%o%y)%o%(x%o%x) - x%o%(y%o%(x%o%x))))
}
X <- cprod(romat())
Y <- cprod(romat())  # X,Y: 6x6 Hermitian quaternionic matrices
ji(X,Y)

thus the Jordan identity is satisfied, to numerical precision. Now the same for 3x3 octonionic matrices:

X <- cprod(romat("octonion",5,3))
Y <- cprod(romat("octonion",5,3))# X,Y: 3x3  Hermitian octonion matrices
ji(X,Y)

but

X <- cprod(romat("octonion",5,4))
Y <- cprod(romat("octonion",5,4))# X,Y: 4x4  Hermitian octonion matrices
ji(X,Y)

showing that 4x4 octonionic Hermitian matrices do not satisfy the Jordan identity, even approximately.

jordan documentation built on April 8, 2021, 5:06 p.m.