研究模李超代数和Hom-型代数的表示与分类，是李理论重要的基础性工作之一，其结果对Hom-Yang-Baxter方程、量子群、共形场论、同调理论、代数表示论等有重要的理论意义。模李超代数(即素特征域上的李超代数)的研究始于20世纪90年代，虽然已有较多研究成果，但有限维单模李超代数的分类尚未完成，并且它的表示也不完善。Hom-型代数的研究始于21世纪，它是将定义代数的等式通过一个线性映射(称为扭曲映射)进行扭曲得到一类更广的代数，本质上是该代数的某种形变。本课题将研究Cartan型单模李超代数、限制单李超代数、有限维单模李超代数的表示与分类，构造新的单模李超代数，从而为单模李超代数的分类提供新方法与例子。同时，还将研究与Hom-Yang-Baxter方程密切相关的Hom-李三系和Hom-左对称代数的表示与分类，应用研究成果将获得Hom-Yang-Baxter方程的新解。

It is one of the important fundamental topics in Lie theory to study representations and classifications of modular Lie superalgebras and Hom-type algebras, and results of this project are important to the study of many subjects such as Hom-Yang-Baxter equation, quantum group, conformal field theory, cohomology theory and algebraic representation theory etc. Modular Lie superalgebras (i.e., Lie superalgebras over a field of prime characteristic) were initiated in the 1990s. Even though many results of modular Lie superalgebras have been obtained, the complete classification of the finite-dimensional simple modular Lie superalgebras remains to be an open problem and its representation theory is not perfect. In twenty-first century, people began to study the Hom-type algebra. Hom-type algebra is a kind of algebras whose some identities are twisted by a linear map (called the twisting map), which can be viewed as a larger class and some deformations of the algebras. This project studies representations and classifications of Cartan-type simple modular Lie superalgebras, restricted simple Lie superalgebras and finite-dimensional simple modular Lie superalgebras, finds some new simple modular Lie superalgebras and provides new methods and examples for the classification of simple modular Lie superalgebras. Moreover, we study the representation and classification of Hom-Lie triple systems and Hom-left symmetric algebras, which are closely related to Hom-Yang-Baxter equation, apply these results to obtain new solutions of Hom-Yang-Baxter equation.