The Frattini subalgebra of \(n\)-Lie algebras (Q2641548)

The study of Frattini subalgebras of Lie algebras has a long history. The present paper considers this idea in \(n\)-Lie algebras. As in Lie algebras, the Frattini subalgebra may not be an ideal as was shown by \textit{Michael Wiliams} [Nilpotent \(n\)-Lie algebras, North Carolina State University, dissertation (2004)] which gives rise to the Frattini ideal, the largest ideal in the algebra which is contained in the Frattini subalgebra. It is shown that if the \(n\)-Lie algebra is nilpotent, then the Frattini subalgebra is an ideal and equals the derived algebra and the Jacobson radical which is the intersection of all maximal ideals.