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.
References
J. H. Conway and D. A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Natick, MA, 2003. MR1957212 1098.17001 J. H. Conway and D. A. Smith, On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Natick, MA, 2003. MR1957212 1098.17001
B. Datta and S. Nag, “Zero-sets of quaternionic and octonionic analytic functions with central coefficients”, Bull. London Math. Soc. 19:4 (1987), 329–336. MR887771 0596.30062 10.1112/blms/19.4.329 B. Datta and S. Nag, “Zero-sets of quaternionic and octonionic analytic functions with central coefficients”, Bull. London Math. Soc. 19:4 (1987), 329–336. MR887771 0596.30062 10.1112/blms/19.4.329
T. Dray and C. A. Manogue, The geometry of the octonions, World Scientific, Hackensack, NJ, 2015. MR3361898 1333.17004 T. Dray and C. A. Manogue, The geometry of the octonions, World Scientific, Hackensack, NJ, 2015. MR3361898 1333.17004
A. Hurwitz, “Über die Komposition der quadratischen Formen”, Math. Ann. 88:1-2 (1922), 1–25. MR1512117 48.1164.03 10.1007/BF01448439 A. Hurwitz, “Über die Komposition der quadratischen Formen”, Math. Ann. 88:1-2 (1922), 1–25. MR1512117 48.1164.03 10.1007/BF01448439
C. A. Manogue and J. Schray, “Finite Lorentz transformations, automorphisms, and division algebras”, J. Math. Phys. 34:8 (1993), 3746–3767. MR1230549 0797.53075 10.1063/1.530056 C. A. Manogue and J. Schray, “Finite Lorentz transformations, automorphisms, and division algebras”, J. Math. Phys. 34:8 (1993), 3746–3767. MR1230549 0797.53075 10.1063/1.530056
A. Putnam, Classifying octonionic-linear operators, master's thesis, Oregon State University, 2017, http://ir.library.oregonstate.edu/xmlui/handle/1957/61707. http://ir.library.oregonstate.edu/xmlui/handle/1957/61707 A. Putnam, Classifying octonionic-linear operators, master's thesis, Oregon State University, 2017, http://ir.library.oregonstate.edu/xmlui/handle/1957/61707. http://ir.library.oregonstate.edu/xmlui/handle/1957/61707
H. Rodríguez-Ordóñez, “Homotopy classification of bilinear maps related to octonion polynomial multiplications”, Linear Algebra Appl. 432:12 (2010), 3117–3131. MR2639272 1206.55017 10.1016/j.laa.2010.01.002 H. Rodríguez-Ordóñez, “Homotopy classification of bilinear maps related to octonion polynomial multiplications”, Linear Algebra Appl. 432:12 (2010), 3117–3131. MR2639272 1206.55017 10.1016/j.laa.2010.01.002
R. Serôdio, “On octonionic polynomials”, Adv. Appl. Clifford Algebr. 17:2 (2007), 245–258. MR2321247 1160.17003 10.1007/s00006-007-0026-y R. Serôdio, “On octonionic polynomials”, Adv. Appl. Clifford Algebr. 17:2 (2007), 245–258. MR2321247 1160.17003 10.1007/s00006-007-0026-y
R. Serôdio, “Construction of octonionic polynomials”, Adv. Appl. Clifford Algebr. 20:1 (2010), 155–178. MR2601896 1222.11136 10.1007/s00006-008-0140-5 R. Serôdio, “Construction of octonionic polynomials”, Adv. Appl. Clifford Algebr. 20:1 (2010), 155–178. MR2601896 1222.11136 10.1007/s00006-008-0140-5
