Construction And Bimodule Of Hom-pre-lie Algebra

Hom-pre-Lie algebras are Hom-type algebras that can be obtained from pre-Lie algebras by twisting along an algebra homomorphism.Hom-pre-Lie algebra is closely related to pre-Lie algebra and it is a special kind of Hom-G-associative algebra.With the development of Lie algebras,the theory of bialgebra has also been widely used in various fields.Hom-pre-Lie bialgebra is a new type of bialgebra based on pre-Lie bialgebra and has many properties similar to the Lie bialgebra.First of all,in this paper we introduce the background about pre-Lie algebras and bialgebras and the development of the theory of bialgebra.This paper mainly inciudes the following three parts:In the first part we give the basic definition of the Hom-Novikov algebras and Hom-pre-Lie algebras.We give several methods for constructing Hom-pre-Lie algebras,for example: the construction of pre-Lie algebra on involutive Hom-pre-Lie algebra and the construction of Hom-pre-Lie algebra on the direct sum of Hom-pre-Lie algebra.In the second part we give the definition of bimodules of Hom-pre-Lie algebra and some basic properties related to it.At the same time,we introduce several special type of bimodules,the left and right multiplication and so on.At last,we give the definition of the matched pairs of Hom-pre-Lie algebras and find the relations between the bimodule of Hom-Lie algebras and Hom-pre-Lie algebras.We also give the definition of compatible representation.At last,we give the definition of Hom-pre-Lie bialgebras and the equivalent condition for an algebra to be a Hom-pre-Lie bialgebra.In the third part of the paper,we obtain some examples of the Hom-Novikov algebras and Hom-pre-Lie algebras for a special endomorphism of Hom-Novikov algebra dimension two.