On Okuba algebras (Q2784300)

In this paper a class of algebras introduced by \textit{S. Okubo} [Hadronic J. 1, No. 4, 1250-1278 (1970; Zbl 0417.17011)] is investigated. Okubo starts from the algebra of \(3\times 3\) matrices over a field \(F\) (of characteristic not \(2\) or \(3\)) and defines a new product on the trace zero elements. The resulting algebra satisfies a composition law, but is in general not unital. (For related work see papers by \textit{T. A. Springer} [Nederl. Akad. Wet., Proc., Ser A 62, 254-264 (1959; Zbl 0092.03701)] and \textit{J. R. Faulkner} [Proc. Am. Math. Soc. 104, No. 4, 1027-1030 (1988; Zbl 0698.17004)].) The authors define an Okubo algebra over a field by the property that a suitable base field extension can be obtained from Okubo's construction. Their main result is a description of Okubo algebras over an arbitrary field \(F\) in terms of central simple associative algebras [resp. simple associative algebras] of the second kind. Furthermore, they show that two Okubo algebras are isomorphic if and only if the Lie algebras defined by the commutator product are isomorphic. (These Lie algebras are simple of type \(A_2\), which is crucial in the proof of the main theorem.) The last section contains a brief discussion of applications.NEWLINENEWLINEFor the entire collection see [Zbl 0984.00502].