Extensions Of Alternative Algebra And Hom-alternative Algebra

In this paper,we mainly study alternative algebras and Hom-alternative algebras,and give their T?-extensions and 7?*-extensions respectively.At the same time,we also study low-dimensional alternative algebras.For a class of four-dimensional non-associative alternative algebra,we calculate the classification of Rota-Baxter operators on them,and find the solutions of alternative Yang-Baxter equation on a special class of eight-dimensional alternative algebra.In the first part,we first introduce some basic concepts of alternative algebra and some conclusions related to this paper.Then we give a concrete method of T--extension of alternative algebra,and prove that T?*-extension of solvable and nilpotent alternative algebra is solvable or nilpotent.Finally,we construct a special T?-extension by linear mapping from alternative algebra to its representation space.In the second part,we give the method to T?-extension of Hom-alternative algebra,construct a special T?-extension by using the special linear mapping from alternative algebra to its representation space,and discuss the case when Hom-alternative algebra is solvable or nilpotent,and prove that the corresponding T?*-extension is solvable or nilpotent.In the third part,we first introduce the definition of Rota-Baxter operator on alternative algebra,the definition of bimodules of alternative algebra and the definition of alternative Yang-Baxter equation.By calculating the matrix of the Rota-Baxter operator on a set of basis,we can calculate the complete classification of Rota-Baxter operators on a class of four-dimensional non-associative alternative algebra,totaling 45 categories.In the fourth part,we first determine the algebraic operations on a class of eight-dimensional alternative algebra corresponding to four-dimensional non-associative alternative algebra,and then use the results of the third part to calculate the solutions of alternative Yang-Baxter equation in the corresponding eight-dimensional alternative algebra for each of the 45 Rota-Baxter operators.In the fifth part,the Rota-Baxter operators on the first kind of four-dimensional alternative algebra calculated in the third part are used to construct pre-alternative algebras.Then,by using the relationship between alternative algebras and pre-alternative algebras,the alternative algebras on the newly constructed pre-alternative algebras are found,and whether the constructed alternative algebras are associative algebras is verified one by one.