Representation Of Compatible Jordan Algebra And BiHom-Jordan Algebra

In this paper,we mainly study BiHom-Jordan algebra and compatible Jordan algebra.we study the representation of BiHom-Jordan algebra and BiHom-pre-Jordan algebra.At the same time,we also study the representation of compatible Jordan algebra and construction of compatible pre-Jordan algebra.Moreover,we also study the construction of Rota-Baxter operators on low-dimensional compatible Jordan algebras.In the first part,we mainly introduce several basic concepts and conclusions related to this paper,including the definition of Jordan algebra and pre-Jordan algebra,the representation of Jordan algebra,its dual representation,its(46)-operator and Rota-Baxter operators.In the meantime,we introduce the classification of Jordan algebra in low dimensions.In the second part,we mainly study the representation of BiHom-Jordan algebra.First,we give the definition of BiHom-Jordan algebra and its equivalent form,and then we give the definition of representation of BiHom-Jordan algebra and its adjoint representation is used as an example to illustrate this definition.Finally,we get the conditions that the dual mapping of representation is a representation of BiHom-Jordan algebra.In the third part,we mainly study the relationship between BiHom-pre-Jordan algebra and BiHom-Jordan algebra.First of all,we give the definition of BiHom-pre-Jordan algebra,and then use the direct sum space of BiHom-pre-Jordan algebra and a linear space to find the definition method of the representation.Then,we give the definition of(46)-operator and RotaBaxter operator on BiHom-Jordan algebra.Finally,we find the relationship between BiHompre-Jordan algebra and BiHom-Jordan algebra.In the fourth part,we mainly study compatible Jordan algebra.To begin with,we give the definition of compatible Jordan algebra,and then we find the conditions that two Jordan algebras are compatible.We also give the definition of representation of compatible Jordan algebra and its adjoint representation is used as an example to illustrate this definition.Then,we get the conditions that the dual mapping of representation is a representation of compatible Jordan algebra.In addition,we discuss compatible pre-Jordan algebra,and find the conditions that two pre-Jordan algebras are compatible.Finally,we give the definition of(46)-operator of compatible Jordan algebra,and find the relationship between compatible Jordan algebra and compatible pre-Jordan algebra.We also find a construction method of compatible pre-Jordan algebra by using compatible Jordan algebra and its(46)-operator.In the fifth part,We mainly construct compatible Jordan algebra in low dimensions,and calculate the classification of Rota-Baxter operators of low-dimensional compatible Jordan algebras.First of all,we specifically discuss whether the two or three-dimensional Jordan algebras are compatible,and then calculate the matrix of its Rota-Baxter operator on the basis of the constructed two-dimensional and three-dimensional compatible Jordan algebras.