The Socle of a Metric n-Lie Algebra

Abstract

We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case.

Key Words:

• Invariant scalar product
• Metric n-Lie algebra
• Socle

2000 Mathematics Subject Classification:

• 17A42
• 17A45
• 17B05

ACKNOWLEDGMENTS

Z. Zhang would like to thank Chern Institute of Mathematics, Nankai University, for its support and hospitality during his visit from October to December in 2009, when this article was completed. C. Bai was supported in part by the National Natural Science Foundation of China (10621101), NKBRPC (2006CB805905), and SRFDP (200800550015).

The authors are very grateful to the referees for their patient and constructive comments, which improved the appearance and readability of this article.

Notes

Communicated by A. Elduque.