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 two such Jordan algebras up to isomorphism. One of them, which was first mentioned by Jordan, Neumann & Wigner (1934) and studied by Albert (1934), is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation

where denotes matrix multiplication. The other is defined the same way, but using split octonions instead of octonions.

Over any algebraically closed field, there is just one Albert algebra. For example, if we complexify the two Albert algebras over the real numbers, we obtain isomorphic Albert algebras over the complex numbers.

The Tits–Koecher construction applied to an Albert algebra gives a form of the E7 Lie algebra.

References

• Albert, A. Adrian (1934), "On a Certain Algebra of Quantum Mechanics", Annals of Mathematics, Second Series 35 (1): 65–73, doi:10.2307/1968118, ISSN 0003-486X, JSTOR 1968118 
• Jordan, P.; Neumann, J. von; Wigner, E. (1934), "On an Algebraic Generalization of the Quantum Mechanical Formalism", Annals of Mathematics (Princeton) 35 (1): 29–64, doi:10.2307/1968117, JSTOR 1968117 
• McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Berlin, New York: Springer-Verlag, doi:10.1007/b97489, ISBN 978-0-387-95447-9, MR2014924, http://books.google.com/books?isbn=0387954473