Cohomology of hemistrict Lie 2-algebras

Abstract

We study representations of hemistrict Lie 2-algebras and give a functorial construction of their cohomology. We prove that both the cohomology of an injective hemistrict Lie 2-algebra L and the cohomology of the semistrict Lie 2-algebra obtained from skew-symmetrization of L are isomorphic to the Chevalley-Eilenberg cohomology of the induced Lie algebra $L_{\text{Lie}}.$

Communicated by Jason P. Bell

Keywords:

• Cohomology
• Leibniz algebras
• Lie 2-algebras

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B56
• 17A32
• 17B70

Acknowledgments

We would like to express our gratitude to several institutions for their hospitality while we were working on this project: Chern Institute of Mathematics (Xiang), Henan Normal University (Cai and Xiang), Peking University (Cai and Xiang), Tsinghua University (Cai and Xiang). We would also like to thank Chengming Bai, Zhuo Chen, Yunhe Sheng, Rong Tang, and Tao Zhang for helpful discussions and comments. Special thanks go to Zhuo Chen for constructive suggestions on this manuscript. We are grateful to the anonymous referee for carefully reading this article and providing us valuable suggestions.

Additional information

Funding

Research partially supported by NSFC grants 11425104, 11901221, and 11931009.