Compatible BiHom-Lie Algebras And Compatible BiHom-pre-Lie Algebras

In this thesis,we mainly study compatible BiHom-Lie algebras and compatible BiHompre-Lie algebras.For these two kinds of algebras,we study the representation and bimodule separately,and study the relationship between the two kinds of algebras and the compatible BiHom-L-dendriform algebras.The first part mainly introduces some basic concepts and relevant conclusions of the text.The second part mainly studies the compatible BiHom-Lie algebras.We first consider the conditions that the linear combination of the two BiHom-Lie algebras is also the BiHomLie algebra,and introduce the concept of compatibility between two BiHom-Lie algebras.Secondly,we consider the method to construct compatible BiHom-Lie algebras in larger spaces based on the given compatible BiHom-Lie algebras and their representations.Similar to the coadjoint representation of Lie algebras,we consider the dual mapping of the representation of compatible BiHom-Lie algebras,computing the conditions that it is also the representation of BiHom-Lie algebras,and find a new class of representation.Finally,the deformation of BiHom-Lie algebras is constructed.The third part mainly studies the compatible BiHom-pre-Lie algebras.Firstly,we find a way to judge the linear combination of algebraic operations of two BiHom-pre-Lie algebras that still gives a BiHom-pre-Lie algebra.Secondly,for the new BiHom-pre-Lie algebras constructed by this method,we discuss how to construct their bimodules by using the primary BiHom-pre-Lie algebras’ bimodules,and find a concrete example.Then we discuss the dual mapping of bimodules of compatible BiHom-pre-Lie algebras,study the conditions that they are also the bimodules of the compatible BiHom-pre-Lie algebras for three different conditions,and construct the new bimodule.Thirdly,we study how to construct compatible BiHom-Lie algebras with compatible BiHom-pre-Lie algebras,and how to construct compatible BiHom-pre-Lie algebras with O-operator of compatible BiHom-Lie algebras.We give a method to get some special representations of the sub-adjacent compatible BiHom-Lie algebras from the bimodule of compatible BiHom-pre-Lie algebras.Finally,we construct the deformation of BiHom-pre-Lie algebras.In the fourth part,we focus on compatible BiHom-L-dendriform algebras.The definition of compatible BiHom-L-dendriform algebra,as well as its bimodule are given.Then we construct two new compatible BiHom-pre-Lie algebras based on one compatible BiHom-L-dendriform algebras,at the same time we find the bimodule of the two compatible BiHom-pre-Lie algebras from the bimodule of the original compatible BiHom-L-dendriform algebras.Finally,we construct the new compatible BiHom-L-dendriform algebras on the representation space of relevant representation of O-operator of compatible BiHom-pre-Lie algebras.