Hom-alternative Bialgebra And Rota-baxter Operators On Low Dimensional Lie Superalgebras

In recent years,with the continuous improvement and development of algebraic theory,the theoretical research on Hom-algebra has received extensive attention from scholars at home and abroad.As an important non-associative algebra,the research on Hom-alternative algebras becomes an important branch.Rota-Baxter operator on Lie algebra is the solution of the classical Yang-Baxter equation of the operator form.This paper mainly studies the construction of the Hom-alternative bialgebra and Rota-Baxter operators on the Lie superalgebras in low dimension.The main structure of this paper is as follows.In the first part,we will introduce some basic concepts of alternative algebra and Hom-a-lternative algebra,and we find the definition of Hom-alternative algebra.We can find the bimodule definition of Hom-alternative algebra by the concepts of alternative algebra.we also find the method of constructing Hom-alternative algebra on the driect sum space of Hom-alternative algebras.And we give a concrete example of the bimodules on Hom-alternative algebra.We find conditions that bimodules of Hom-alternative algebra is still bimodules of Hom-alternative algebra.We get the conditions of compatible Hom-alternative algebra.In the second part,we give the definition of mathed pairs on Hom-alternative algebra and we obtain a method that constructs Hom-alternative algebras on dirct the sum of two Hom-alternative algebras.In the third part,the definition and method of constructing Hom-alternative bialgebra are given.The method of constructing Hom-alternative algebra on the dirct sum of the Hom-alte-rnative algebra and its duality of Hom-alternative algebra is found.And we get the equivalent condition of constructing Hom-alternative bialgebra.In the fourth part,we use the classification of two-dimensional Lie superalgebras to calculate Rota-Baxter operators on two kinds of two-dimensional Lie superalgebras.In the fifth part,we use the classification of three-dimensional Lie superalgebras and calculate the four Rota-Baxter operators on four kinds of three-dimensional Lie superalgebras.We obtain classificatis on Rota-Baxter operators on three-dimensional Lie superalgebras.In the sixth part we use the relation between Rota-Baxter operators on Lie superalgebras and the solution of the classical Yang-Baxter equation on the direct sum of the Lie superalgebra and the dual space,and for the Rota-Baxter operators in the fifth part,we find the solution of the super classical Yang-Baxter equation in term forms in g(?)g*.