Monoidal Hom-Hopf代数是近年来Hopf代数研究的新方向，其上的自同态Hom-代数的结构和扩张是重要内容，其中monoidal范畴和积分理论是有效工具。. 本项目的关键是：构造自同态空间合理的Hom-余模代数结构，并基于Hom-Hopf模范畴的基本结构定理，建立它和余不变Hom-子代数的Hom-smash积之间同构关系；由Hom-积分诱导Hom-余模形式的Hom-函子和张量积函子之间的伴随对关系；由对称的Markov扩张和depth 2扩张，定义中心化子的monoidal Hom-Hopf代数结构，并构造Jones tower中自同态Hom-代数的Hom-Hopf Galois扩张。. 分三方面研究：Hom-余模代数的结构刻画；自同态Hom-代数的结构和对偶定理；以及Hom-Hopf Galois扩张和depth 2扩张，以及Hochschild上同调等问题。

Monoidal Hom-Hopf algebras have become a new direction of the study of Hopf algebras, over which the algebraic structures and extensions of endomorphism Hom-algebras are the main research contents. The monoidal category and integral theory are effective instruments. The following are the key point for this project:. Constructing a reasonable Hom-comodule algebraic structure on the endomorphism spaces, and the isomorphism as Hom-algebras between Hom-smash products of their coinvariant Hom-subalgebras and them, based on the fundamental structure theorem of the category of Hom-Hopf modules; Inducing Hom-comodule type of Hom-functor and tensor product functor such that form an adjoint pair by Hom-integrals; Constructing a monoidal Hom-Hopf algebra structure on the centralizer induced by symmetric Markov extension and depth 2 extension, and Hom-Hopf Galois extension of the endomorphism Hom-algebra in a Jones tower.. Concretely, we will do in three parts: characterization of the structure of Hom-comodule algebras; the structure and duality theorems of endomorphism Hom-algebras; extension and Hochschild cohomology problems of Hom-Hopf Galois and depth 2.