Hom-Type (Co) Algebra Structure And Nonlinear Hom-Type Equations

In recent years, Hom-type algebra structure become one of the important research topics and attracted more and more attention of scholars. Hom-(co)algebras is actual-ly a generalization of (co)algebra. Here the (co)associativity is replaced by the Hom-(co)associativity where α:A → A(C → C) is a linear map.Based on the structures of Hom-type algebra, we focus on some non-linear equations, such as:Hom-Hopf equation, Hom-Pentagon equation, Hom-Frobenius-separability equa-tion and Horn-Yang-Baxter equation. The relations between the solutions of non-linear equations and the structures of Hom-type (co)algebras are discussed. By using the struc-tures of Hom-type (co)algebras, some solutions of these equations are constructed. The main results are as follows:(1) Some preliminary definitions and basic results are given. Hom-(co)algebra, Hom-bialgebra, Hom-(co)module and Hom-bi(co)module are included.(2) A discussion of Hom-Hopf(Pentagon) equation and Hom-Hopf algebra eleinen-t(map) is provided. We discuss how the Hom-Hopf algebra element(map) can be used to construct the solutions of Hom-Hopf(Pentagon) equation.(3) For Hom-Frobenius-separability equation, we see that all solutions of which are also solutions of the Hom-braid equation. It is shown that any Hom-central element is a solution of Hom-Frobenius-separability equation. Dually, any Hom-FS map is also a solution of Hom-Frobenius-separability equation.(4) A method to construct solutions of Hom-Yang-Baxter equations is presented. Thus we can get a so called a-involutory solution of Horn-Yang-Baxter equation from every Hom-algebras structure on a space. The converse of it is considered as well.