Some results on metric n-Lie algebras

Abstract

We study the structure of a metric n-Lie algebra \(\mathcal{G}\) over the complex field 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 \(m\left( \mathcal{G} \right)\) 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 \({\mathcal{G} \mathord{\left/ {\vphantom {\mathcal{G} \mathcal{R}}} \right. \kern-\nulldelimiterspace} \mathcal{R}}\). 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 \(\mathcal{R}^ \bot \subseteq \mathcal{R}\).