Capability of Crossed Modules of Lie Algebras

Abstract

In this paper, we introduce the concept of capability for crossed modules of Lie algebras, which is a generalization of capability in Lie algebras and groups. By using a special central ideal of a crossed module, we give a sufficient condition for the capability of a crossed module of Lie algebras. Also, we will extend the five-term exact sequence on homology of crossed modules of Lie algebras one term further and study the connection between the capability of crossed modules and this sequence. Finally, we study the relation between the capability and the center of a cover of a crossed module.

Key Words:

• Capability
• Cover of crossed modules
• Crossed module of Lie algebras
• Homology of crossed modules of Lie algebras

2010 AMS Subject Classification:

• 17B56
• 17B99
• 18G99

ACKNOWLEDGMENT

The author would like to thank the referee for his/her interesting comments and suggestions that helped improve the clarity and the relevance of this paper.

Notes

Communicated by A. Elduque.