Metric $n$-Lie Algebras

TL;DR

This paper investigates the structure of metric $n$-Lie algebras over complex numbers, focusing on their Levi decomposition, minimal ideals, and invariant forms, revealing new structural relationships and conditions for semisimplicity.

Contribution

It provides new structural insights into metric $n$-Lie algebras, including relationships between radicals, minimal ideals, and invariant forms, and characterizes conditions for the absence of strong semisimple ideals.

Findings

$ ext{dim}( ext{space of invariant forms} \\geq m( ext{G}) + 1$

$ ext{centralizer of R} = ext{sum of minimal ideals}$

Condition for no strong semisimple ideals: $ ext{R}^ot \\subseteq ext{R}$

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 $\mathcal R^\bot \subseteq \mathcal R$.