Eight Kinds Of 3-dimensional Jordan Algebras And Its Generalization

In 1930s P.Jordan first gave the definition of Jordan algebra.Jordan algebra is a very important non-associative algebra,it is widely used in many fields.In this paper we mainly discuss the Rota-Baxter operators on three-dimensional Jordan algebras.We also give the definitions of Hom-pre-Jordan algebras?Hom-J-dendriform algebras and Hom-J-quadri algebras and the relationship between these algebras.In section one we introduce some basic definitions,including the definition of Jordan algebra?the representation of Jordan algebra?J-dendriform algebra?J-quadri algebra?Rota-Baxter operator and so on.We also give some basic conclusions.In section two we calculate the Rota-Baxter operators on eight kinds of three-dimensional Jordan algebras.First,we find the classification of Jordan algebras in two and three dimension,then we calculate the Rota-Baxter operators on eight kinds of three-dimensional Jordan algebras and get the complete classification of Rota-Baxter operators.In section three we mainly discuss the solutions of Jordan Yang-Baxter equation on six-dimensional Jordan algebrasJa_rg?9?J^?9?.First,we determine the algebraic operation of six-dimensional Jordan algebrasJa_rg?9?J^?9?,then we use the Rota-Baxter operators of the Jordan algebras to get the solutions of Jordan Yang-Baxter equation inJa_rg?9?J^?9?.In section four we use the Rota-Baxter operators on three-dimensional Jordan algebras to construct pre-Jordan algebras on Jordan algebras,we also determine the Jordan algebras on pre-Jordan algebras.In section five we discuss the generalization of Jordan algebras.First,we introduce the definitions of Hom-pre-Jordan algebras?Hom-J-dendriform algebras and Hom-J-quadri algebras,then we discuss the relationship among pre-Jordan algebras and Hom-pre-Jordan algebras?J-dendriform algebras and Hom-J-dendriform algebras?J-quadri algebras and Hom-J-quadri algebras,finally,we discuss the relationship among Hom-Jordan algebras?Hom-pre-Jordan algebras?Hom-J-dendriform algebras and Hom-J-quadri algebras.