Extensions Of 3-Pre-Lie Algebras

This paper mainly studies 3-pre-Lie algebras,defines its matched pair and generalized representation,and then constructs three different extensions of 3-pre-Lie algebras.At the same time,this paper also studies the hom-3-pre-Lie algebras,defines the representation of multiplicative hom-3-pre-Lie algebras and studies the dual mapping of its representation.The first part mainly introduces the basic concepts related to this paper,including the definition of 3-Lie algebras,the representation of 3-Lie algebras and its dual representation,the definition of 3-pre-Lie algebras,the representation of 3-pre-Lie algebras and its dual representation.The second part defines the matched pair of 3-pre-Lie algebras.For any two 3-pre-Lie algebras,a new 3-pre-Lie algebras is constructed on their direct sum by using their special representation.The third part defines the generalized representation of 3-pre-Lie algebras.For the direct sum of 3-pre-Lie algebras and its representation space,the direct sum space can be regarded as 3-pre-Lie algebras by defining new algebraic operations.In the fourth part,we define the non-abelian extensions of 3-pre-Lie algebras,and find a new method to construct 3-pre-Lie algebras from the direct sum of two 3-pre-Lie algebras.In the fifth part,we first define the 3-cocycle of 3-pre-Lie algebras with respect to representation,define another operation on the direct sum of 3-pre-Lie algebras and its representation space,and find the equivalent conditions for the direct sum space to form 3-pre-Lie algebras for this operation.Then,we give the concept of Tθ-extensions of 3-pre-Lie algebras,give the special Tθ-extensions method,and discuss the isomorphic relation between two kinds of special Tθ-extensions.In the sixth part,we first define the Tθ*-extensions of 3-pre-Lie algebras,and prove that the Tθ*-extensions of solvable and nilpotent 3-pre-Lie algebras is solvable or nilpotent.Finally,a new bilinear function is defined on the extension of 3-pre-Lie algebras.In the seventh part,we study hom-3-pre-Lie algebras.Firstly,we define new algebraic operations to guarantee that the direct sum space of two hom-3-pre-Lie algebras is hom-3-pre-Lie algebras.Then,we study the necessary and sufficient condition that the linear mapping between Hom-3-pre-Lie algebras is homomorphic mapping.Finally,we define the representation of hom-3-pre-Lie algebras and compute condition for the dual of mapping to be a representation.