Extension Theory Of Pre-Lie Algebras

In this paper,we study the extension theory of pre-Lie algebras.First,we show that non-abelian extensions of pre-Lie algebras can be classified by the non-abelian cohomology.Then we naturally construct a strict Lie 2algebras by the bimultipliers of pre-Lie algebras.We show that a non-abelian extension of pre-Lie algebras naturally gives rise to a homomorphism from the subadjacent Lie algebra to the strict Lie 2-algebra constructed from the bimultipliers,and isomorphism classes of non-abelian extensions of pre-Lie algebras one-to-one correspond to homotopy classes of the aforementioned homomorphisms.Finally,we classify the non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements in a differential graded Lie algebra.We give a concrete example to compute the non-abelian cohomology of pre-Lie algebras.The paper is divided into six chapters.In Chapter 1,we introduce the background and progress of the research topic,and then introduce the research motivation and main results.In Chapter 2,we review the list of Lie algebras,pre-Lie algebras,subadjacent Lie algebras and pre-Lie algebras.The concepts of central extension and commutative extension of pre-Lie algebras are given,and it is proved that(g ⊕ η,＊（ρ,μ,ω))is a pre-Lie algebras if and only if(η;ρ,μ)is the representation of g,and ω is a 2-cocycle on the representation.In Chapter 3,we give the definition of non-abelian extensions of preLie algebras and define the non-abelian cohomology for pre-Lie algebras and show that non-abelian extensions of pre-Lie algebras can be classified by the non-abelian cohomology.In Chapter 4,we define the bimultipliers for a pre-Lie algebra.In addition,we construct a strict Lie 2-algebra from the bimultipliers for a pre-Lie algebra.We show that a non-abelian extension of pre-Lie algebras naturally gives rise to a homomorphism from the subadjacent Lie algebra to the strict Lie 2-algebra constructed from the bimultipliers,and isomorphism classes of non-abelian extensions of pre-Lie algebras one-to-one correspond to homotopy classes of the aforementioned homomorphisms.In Chapter 5,we classify the non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements in a differential graded Lie algebra.In Chapter 6,we make a brief summary of this paper and give a concrete example to compute the non-abelian cohomology of pre-Lie algebras.