Loterre: Mathématiques: Albert algebra

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Albert algebra

In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism. One of them, which was first mentioned by Pascual Jordan, John von Neumann, and Eugene Wigner (1934) and studied by Albert (1934), is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation
${\\displaystyle x\\circ y={\ rac {1}{2}}(x\\cdot y+y\\cdot x),}$

where ⋅ denotes matrix multiplication. Another is defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Albert_algebra)

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