Representation And Bialgebra Structure Of Compatible Malcev Algebras

In this thesis,we mainly study compatible Malcev algebras and compatible pre-Malcev algebras,and the construction methods of compatible Malcev bialgebras.In chapter 1,we introduce the basic concepts and related results of Malcev algebras and pre-Malcev algebras.In chapter 2,we study the conditions that two Malcev algebras are compatible and the representation of compatible Malcev algebras.Firstly,we find the conditions that the linear combination of two Malcev algebras is still Malcev algebras by concrete calculations.Then,we consider the conditions that the linear combination of the representations of two compatible Malcev algebras is the representation of Malcev algebras,construct its a special representation,and demonstrate the existence and rationality of the definition by concrete examples.We further construct the examples of the representation of compatible Malcev algebras by considering the dual mapping of the representation of compatible Malcev algebras.We extend the definition of-operator of compatible Lie algebras to compatible Malcev algebras,and obtain the equivalent conditions for this definition.In chapter 3,we study the relationship between compatible Malcev algebras and compatible pre-Malcev algebras in terms of the algebraic structure and the bimodule.Firstly,we find the conditions that two pre-Malcev algebras are compatible and the bimodules of two compatible pre-Malcev algebras are compatible,and an example of special bimodule that satisfy these conditions.On the one hand,we construct compatible Malcev algebras based on compatible pre-Malcev algebras by constructing special algebraic operations,on the other hand,for compatible pre-Malcev algebras and compatible Malcev algebras derived from it,we discuss specifically the intrinsic relation between their representations or bimodules.In addition,combining the definition of-operator in chapter 2,we construct two compatible pre-Malcev algebras based on the representation space of compatible Malcev algebras.In chapter 4,we study the construction methods of compatible Malcev bialgebras.Firstly,starting with two compatible Malcev algebras,we calculate the conditions that there are two compatible Malcev algebras on their direct sum space by constructing special representations.Then,we study how to construct two compatible Malcev algebraic structures on the dual space of compatible Malcev algebras by the dual mapping of two special linear mappings on compatible Malcev algebras.We extend the definition of the Manin triple and standard Manin triple of Lie algebras to compatible Malcev algebras,and discuss the relationship between compatible Malcev bialgebras,the standard Manin triple of compatible Malcev algebras and the matched pair of compatible Malcev algebras.