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Octonion associators

Sangwine, Stephen J (2015) Octonion associators. Working Paper. arXiv. (Unpublished)

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Abstract

The algebra of octonions is non-associative (as well as non-commutative). This makes it very difficult to derive algebraic results, and to perform computation with octonions. Given a product of more than two octonions, in general, the order of evaluation of the product (placement of parentheses) affects the result. Inspired by the concept of the commutator $[x,y]=x^{-1}y^{-1}xy$ we show that an associator can be defined that multiplies the result from one evaluation order to give the result from a different evaluation order. For example, for the case of three arbitrary octonions $x$, $y$ and $z$ we have $((xy)z)a = x(yz)$, where $a$ is the associator in this case. For completeness, we include other definitions of the commutator, $[x,y]=xy-yx$ and associator $[x,y,z]=(xy)z-x(yz)$, which are well known, although not particularly useful as algebraic tools. We conclude the paper by showing how to extend the concept of the multiplicative associator to products of four or more octonions, where the number of evaluation orders is greater than two.

Item Type: Monograph (Working Paper)
Uncontrolled Keywords: math.RA, math.RA, 17A35
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health > Computer Science and Electronic Engineering, School of
Depositing User: Jim Jamieson
Date Deposited: 13 Dec 2016 22:15
Last Modified: 21 Sep 2018 16:15
URI: http://repository.essex.ac.uk/id/eprint/18547

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