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 $m\left( \mathcal{G} \right)$ 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 $m\left( \mathcal{G} \right) + 1$ . 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 $\mathcal{R}^ \bot \subseteq \mathcal{R}$ .