Noncommutative Jordan algebra

In algebra, a noncommutative Jordan algebra is an algebra, usually over a field of characteristic not 2, such that the four operations of left and right multiplication by x and x² all commute with each other. Examples include associative algebras and Jordan algebras.

Over fields of characteristic not 2, noncommutative Jordan algebras are the same as flexible Jordan-admissible algebras, where a Jordan-admissible algebra – introduced by Albert (1948) and named after Pascual Jordan – is a (possibly non-associative) algebra that becomes a Jordan algebra under the product a ∘ b = ab + ba.

See also

• Malcev-admissible algebra
• Lie-admissible algebra

References

• Albert, A. Adrian (1948), "Power-associative rings", Transactions of the American Mathematical Society, 64 (3): 552–593, doi:10.2307/1990399, JSTOR 1990399, MR 0027750
• Okubo, Susumu (1995), Introduction to octonion and other non-associative algebras in physics, Montroll Memorial Lecture Series in Mathematical Physics, vol. 2, Cambridge: Cambridge University Press, ISBN 0-521-47215-6, Zbl 0841.17001
• Schafer, R. D. (1955), "Noncommutative Jordan algebras of characteristic 0", Proc. Amer. Math. Soc., 6 (3): 472–5, doi:10.1090/s0002-9939-1955-0070627-0, JSTOR 2032791, MR 0070627