Vol. 12, No. 1, 2019

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Classifying linear operators over the octonions

Alex Putnam and Tevian Dray

Vol. 12 (2019), No. 1, 117–124

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.

octonions, quaternions, division algebras, linear operators, linear equations
Mathematical Subject Classification 2010
Primary: 17A35
Secondary: 15A06
Received: 23 July 2017
Revised: 4 January 2018
Accepted: 14 February 2018
Published: 31 May 2018

Communicated by Jim Hoste
Alex Putnam
Department of Mathematics
Oregon State University
Corvallis, OR
United States
Tevian Dray
Department of Mathematics
Oregon State University
Corvallis, OR
United States