Leibniz 2-Algebras and Twisted Courant Algebroids

Abstract

In this article, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in [26] as well as twisted Courant algebroids by closed 4-forms introduced in [13]. We also prove that Dirac structures of twisted Courant algebroids give rise to 2-term L _∞-algebras and geometric structures behind them are exactly H-twisted Lie algebroids introduced in [10].

Key Words:

• Dirac structure
• Leibniz 2-algebra
• Omni-Lie 2-algebras
• The first Pontryagin class
• Twisted Courant algebroid

2010 Mathematics Subject Classification:

• 17B99
• 53D17

ACKNOWLEDGMENTS

We would like to thank P. Bouwknegt, M. Grützmann, Sen Hu, Bailing Wang, and Chenchang Zhu for helpful discussions and comments. We also give our special thanks to referees for many helpful suggestions.

Research was supported by NSFC (11101179, 11026046), Doctoral Fund. of MEC (20090001110006, 20100061120096), and “the Fundamental Research Funds for the Central Universities” (200903294).

Notes

Communicated by A. Elduque.