Non-abelian extensions of pre-Lie algebras

Abstract

In this paper, we study non-abelian extensions of pre-Lie algebras. First we introduce the non-abelian cohomology for pre-Lie algebras by which we classify non-abelian extensions of pre-Lie algebras. Then we construct the bimultipliers for pre-Lie algebras, which naturally gives rise to a strict Lie 2-algebra. We show that there is a one-to-one correspondence between isomorphic classes of non-abelian extensions of pre-Lie algebras and homotopy classes of homomorphisms from the subadjacent Lie algebra to the strict Lie 2-algebra obtained from the bimultipliers. Finally we classify non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements of a differential graded Lie algebra.

Keywords:

• Maurer-Cartan element
• Lie 2-algebra
• non-abelian cohomology
• non-abelian extension
• pre-Lie algebra

2020 Mathematics Subject Classification:

• 17A30
• 17B99

Acknowledgments

We give warmest thanks to the referee for helpful suggestions.

Additional information

Funding

This research is supported by NSFC (12001226).