Bimodules Of BiHom-pre-Lie Algebras

In this paper we study the bimodules of BiHom-pre-Lie algebras.At the same time,we study the representations of BiHom-Lie algebras,the matched pair of BiHom-Lie algebras and BiHom-pre-Lie algebras.The relations among BiHom-pre-Lie algebras,BiHom-L-den driform algebras and BiHom-L-quadri algebras is also discussed.In the first part,we introduce some basic concepts of algebras and related conclusions,which provide theoretical basis for later research.In the second part,we first give the conditions that the dual mapping of the represen tations of BiHom-Lie algebras is also a representations of BiHom-Lie algebras.We get the conditions that the dual mapping of adjoint representations is also a representation of BiHomLie algebras.Secondly,we construct a new kind of dual mapping and we prove it is a repres entations of BiHom-Lie algebra.Finally,we give the conditions for the existence of BiHomLie algebras on the dual space of BiHom-Lie algebras.In the third part,we first give the definition of bimodules of BiHom-pre-Lie algebras,we also give the methods to judge bimodules and an example of bimodules.We discuss the relationship between the bimodules of BiHom-pre-Lie algebras and the representations of BiHom-Lie algebras.Secondly,we show the conditions that the dual mapping of bimodules of BiHom-pre-Lie algebras is still a bimodule,and construct a new kind of dual mapping,and further obtains a kind of bimodules of BiHom-pre-Lie algebras.Finally,this paper gives the conditions for the existence of BiHom-pre-Lie algebras on the dual space of BiHom-pre-Lie algebras.In the fourth part,we first give the definition of the matched pair of BiHom-Lie algebras and give the methods to judge the matched pair of BiHom-Lie algebras.Then we give the definition of the matched pair of BiHom-pre-Lie algebras and find the methods of judging the matched pair of BiHom-pre-Lie algebras.Secondly,we give the relationship between the matched pair of BiHom-Lie algebras and the matched pair of BiHom-pre-Lie algebras.Finally,the conditions for the existence of the structure of BiHom-Lie algebras on the direct sum space of the compatible BiHom-Lie algebras and its dual spaces are given.In the fifth part,we first give the definition of BiHom-L-dendriform algebras and point the relationship between L-dendriform algebras and BiHom-dendriform algebras.Secondly,we give the methods of constructing BiHom-pre-Lie algebras on BiHom-L-dendriform algebras.Finally,we give the relationship between the left and right multiplication operator of BiHom-L-dendriform algebras and the bimodules of BiHom-pre-Lie algebras.In the sixth part,we give the definition of BiHom-L-quadri algebras and give the methods of constructing BiHom-L-quadri algebras on L-quadri algebras.The relationship between BiHom-L-quadri algebras and BiHom-pre-Lie algebras,BiHom-L-dendriform alg ebras,BiHom-dendriform algebras is disscussed.