Module extensions of 3-Lie algebras

Abstract

This article describes module extensions of 3-Lie algebras. In particular, we introduce the notion of T _θ-extensions and -extensions of 3-Lie algebras in terms of modules. We then provide the necessary and sufficient conditions for a 2k-dimensional metric 3-Lie algebra to be isomorphic to a T _θ-extension. We also study properties of the -extension of nilpotent metric 3-Lie algebras. At the end of this article, we obtain all the -extensions of four-dimensional 3-Lie algebras.

Keywords:

• 3-Lie algebra
• module
• T _θ-extension
• T _θ*-extension

AMS Subject Classifications::

• 17B05
• 17D99

Acknowledgement

This project is partially supported by NSF(10871192) of China and NSF(A2010000194) of Hebei Province, China.