Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra $mathcal{G}$ over the complex field ?. Let $mathcal{G} = mathcal{S} oplus mathcal{R}$ be the Levi decomposition, where $mathcal{R}$ is the radical of $mathcal{G}$ and $mathcal{S}$ is a strong semisimple subalgebra of $mathcal{G}$ . Denote by $mleft( mathcal{G} right)$ the number of all minimal ideals of an indecomposable metric n-Lie algebra and $mathcal{R}^ bot$ the orthogonal complement of R. We obtain the following results. As $mathcal{S}$ -modules, $mathcal{R}^ bot$ is isomorphic to the dual module of ${mathcal{G} mathord{left/ {vphantom {mathcal{G} mathcal{R}}} right. kern-0em} mathcal{R}}$ . The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on $mathcal{G}$ is equal to that of the vector space of certain linear transformations on $mathcal{G}$ ; this dimension is greater than or equal to $mleft( mathcal{G} right) + 1$ . The centralizer of $mathcal{R}$ in $mathcal{G}$ is equal to the sum of all minimal ideals; it is the direct sum of $mathcal{R}^ bot$ and the center of $mathcal{G}$ . Finally, $mathcal{G}$ has no strong semisimple ideals if and only if $mathcal{R}^ bot subseteq mathcal{R}$ .