Classifications of (n + k)-dimensional metric n-Lie algebras

We study the structure of isotropic ideals of the metric n-Lie algebra over the complex field for n 2. Based on the study of isotropic ideals of metric n-Lie algebras in this paper, we obtain the following. (i) Let be the Levi decomposition and without semi-simple ideals. If n > 2, is perfect and is isotropic, then is the unique isomaximal ideal and , and the metric dimension of is m. If is indecomposable, then is isomorphic to as an -module in the regular representation. (ii) If with 2 k n + 1, then . If is unsolvable, then is reductive. If the center of is isotropic, then (iii) There are only three classes of (n + k)-dimensional metric n-Lie algebras for 2 k n + 1 and n > 2, and the concrete multiplications for each class are provided.