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Classifying
linear operators over the octonions
Alex Putnam and Tevian Dray
Vol. 12 (2019), No. 1, 117–124
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Abstract |
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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. |
Keywords
octonions, quaternions, division algebras, linear
operators, linear equations
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Mathematical Subject Classification 2010
Primary: 17A35
Secondary: 15A06
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Milestones
Received: 23 July 2017
Revised: 4 January 2018
Accepted: 14 February 2018
Published: 31 May 2018
Communicated by Jim Hoste
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Authors |
| Alex Putnam | |
| Department of Mathematics Oregon State University Corvallis, OR United States |
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| Tevian Dray | |
| Department of Mathematics Oregon State University Corvallis, OR United States |
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