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.