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Cayley Algebra


The only nonassociative division algebra with real scalars. There is an 8-square identity corresponding to this algebra.

The elements of a Cayley algebra are called Cayley numbers or octonions, and the multiplication table for any Cayley algebra over a field F with field characteristic p!=2 may be taken as shown in the following table, where u_1, u_2, ..., u_8 are a bases over F and mu_1, mu_2, and mu_3 are nonzero elements of F (Schafer 1996, pp. 5-6).

u_1u_2u_3u_4u_5u_6u_7u_8
u_1u_1u_2u_3u_4u_5u_6u_7u_8
u_2u_2mu_1u_1-u_4-mu_1u_3-u_6-mu_1u_5u_8mu_1u_7
u_3u_3u_4mu_2u_1mu_2u_2-u_7-u_8-mu_2u_5-mu_2u_6
u_4u_4mu_1u_3-mu_2u_2-mu_1mu_2u_1-u_8-mu_1u_7mu_2u_6mu_1mu_2u_5
u_5u_5u_6u_7u_8mu_3u_1mu_3u_2mu_3u_3mu_3u_4
u_6u_6mu_1u_5u_8mu_1u_7-mu_3u_2-mu_1mu_3u_1-mu_3u_4-mu_1mu_3u_3
u_7u_7-u_8mu_2u_5-mu_2u_6-mu_3u_3mu_3u_4-mu_2mu_3u_1mu_2mu_3u_2
u_8u_8-mu_1u_7mu_2u_6-mu_1mu_2u_5-mu_3u_4mu_1mu_3u_3-mu_2mu_3u_2mu_1mu_2mu_3u_1

See also

Cayley Number, Division Algebra, Octonion, Nonassociative Algebra

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References

Kurosh, A. G. General Algebra. New York: Chelsea, pp. 226-228, 1963.Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, pp. 5-6, 1996.

Referenced on Wolfram|Alpha

Cayley Algebra

Cite this as:

Weisstein, Eric W. "Cayley Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CayleyAlgebra.html

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