Metric n-Lie Algebras

Abstract

An n-Lie algebra is said to be metric if it is endowed with an invariant, non-degenerate, symmetric bilinear form. We prove that any simple n-Lie algebra over an algebraically closed field of characteristic zero admits a unique metric structure and vice versa. Further, we present two metric n-Lie algebras, which are indecomposable but admit many more metric structures.

Key Words:

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

2000 Mathematics Subject Classification:

• 17A42
• 17B20
• 17B81

ACKNOWLEDGMENT

The authors wish to thank the referee for constructive comments which improved the exposition and readability of this article.

This work is supported by the NSF (A2007000138) of Hebei Province.

Notes

Communicated by A. Eldugue.