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2019 Classifying linear operators over the octonions
Alex Putnam, Tevian Dray
Involve 12(1): 117-124 (2019). DOI: 10.2140/involve.2019.12.117

Abstract

We classify linear operators over the octonions and relate them to linear equations with octonionic coefficients and octonionic variables. Along the way, we also classify linear operators over the quaternions, and show how to relate quaternionic and octonionic operators to real matrices. In each case, we construct an explicit basis of linear operators that maps to the canonical (real) matrix basis; in contrast to the complex case, these maps are surjective. Since higher-order polynomials can be reduced to compositions of linear operators, our construction implies that the ring of polynomials in one variable over the octonions is isomorphic to the product of eight copies of the ring of real polynomials in eight variables.

Citation

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Alex Putnam. Tevian Dray. "Classifying linear operators over the octonions." Involve 12 (1) 117 - 124, 2019. https://doi.org/10.2140/involve.2019.12.117

Information

Received: 23 July 2017; Revised: 4 January 2018; Accepted: 14 February 2018; Published: 2019
First available in Project Euclid: 26 October 2018

zbMATH: 06887335
MathSciNet: MR3810482
Digital Object Identifier: 10.2140/involve.2019.12.117

Subjects:
Primary: 17A35
Secondary: 15A06

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 1 • 2019
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