Geodesic
Real Flexible Division Algebras
Canadian journal of mathematics, Tome 34 (1982) no. 3, pp. 550-588

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3].All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A. 
Benkart, Georgia M.; Britten, Daniel J.; Osborn, J. Marshall. Real Flexible Division Algebras. Canadian journal of mathematics, Tome 34 (1982) no. 3, pp. 550-588. doi: 10.4153/CJM-1982-039-x
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[1] 1. Anderson, T., A note on derivations of commutative algebras, Proc. A.M.S. 17 (1966)' 1199–1202. Google Scholar

[2] 2. Benkart, G. M. and Osborn, J. M., The derivation algebra of a real division algebra, Amer. J. Math. 103 (1981), 1135–1150. Google Scholar

[3] 3. Benkart, G. M. and Osborn, J. M., An investigation of real division algebras using derivations, Pacific J. Math. 96 (1981), 265–300. Google Scholar

[4] 4. Benkart, G. M. and Osborn, J. M., Flexible Lie-admissible algebras, J. Algebra 71 (1981), 11–31. Google Scholar

[5] 5. Bruck, R. H., Some results in the theory of linear non-associative algebras, Trans. A.M.S. 56 (1944), 141–199. Google Scholar

[6] 6. Hopf, H., Ein topologischer Beitrag zur reelen Algebra, Commentarii Mathematici Helvetici 13 (1940), 219–239. Google Scholar

[7] 7. Kervaire, M., Non-par alienability of the n sphere for n > 1, Proc. Nat. Acad. Sci. 44 (1958), 280–283. +1,+Proc.+Nat.+Acad.+Sci.+44+(1958),+280–283.>Google Scholar

[8] 8. J., Milnor and R., Bott, On the parallelizability of the spheres, Bull. A.M.S. 64. (1958) 87-89. Google Scholar

[9] 9. Okubo, S., Pseudo-quaternion and pseudo-octonian algebras, Hadronic Journal 1 (1978), 1250–1278. Google Scholar

[10] 10. Okubo, S. and Myung, H. C., Some new classes of division algebras, J. Algebra 67 (1980), 479–490. Google Scholar

[11] 11. Okubo, S. and Osborn, J. M., Algebras with nondegenerate associative symmetric bilinear forms permitting composition, Communications in Algebra 9 (1981), 1233–1261. Google Scholar

[12] 12. Osborn, J. M., Quadratic division algebras, Trans. A.M.S. 105 (1962), 78–92. Google Scholar

[13] 13. Raffin, R., Anneaux a puissances commutative et anneaux flexibles, C. R. Ac. Se. Paris 230 (1950), 804–806. Google Scholar

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