Metric $n$-Lie Algebras

Abstract

We study the structure of a metric $n$-Lie algebra $\mathcal {G}$ over the complex field $\mathbb C$. 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 $m(\mathcal {G})$ 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 $\mathcal {G} / \mathcal R.$ The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on $\mathcal {G}$ equals that of the vector space of certain linear transformations on $\mathcal {G}$; this dimension is greater than or equal to $m(\mathcal {G}) + 1$. The centralizer of $\mathcal R$ in $\mathcal G$ equals the sum of all minimal ideals; it is the direct sum of $\mathcal R^\bot$ and the center of $\mathcal {G}$. The sufficient and necessary condition for $\mathcal {G}$ having no strong semisimple ideals is that $\mathcal R^\bot \subseteq \mathcal R$.