Hom-左对称代数的结构与分类

左对称代数是在理论物理、微分几何及数学物理等领域中起重要作用的一类非结合代数，它与李代数密切相关。而Hom-左对称代数作为左对称代数的推广，与Hom-Yang-Baxter方程密切相关，是研究Hom-李容许代数时引入的代数结构，这是新兴的研究对象，因此还有许多问题亟待研究。. 本项目计划研究Hom-左对称代数的结构与分类，具体内容包括：建立Hom-左对称代数的泛中心扩张理论；给出Hom-左对称代数与李代数之间的联系，再利用Virasoro代数和Witt代数的表示理论对Hom-左对称代数进行分类。特别地，对于一类特殊的Hom左对称代数—Hom-Novikov代数，研究其交换扩张、中心扩张、T*-扩张和形变等问题。

A left-symmetric algebra is a kind of non-associative algebra that plays an important role in the fields of theoretical physics, differential geometry and mathematical physics, which is closely related to Lie algebras. As a generalization of left-symmetric algebra, the notion of Hom-left-symmetric algebras is introduced to study Hom-Lie admissible algebras, which is also closely related to Hom-Yang-Baxter equation. It is an emerging research object, so there are still many problems to be studied..This project aims to study the structure and classification of Hom-left-symmetric algebras. We will study the universal central extension theory of Hom-left-symmetric algebras. We will also establish the relationship between Hom-left-symmetric algebras and Lie algebras, and apply the representation theory of Virasoro algebras and Witt algebras to classify Hom-left-symmetric algebras. Moreover, for a special kind of Hom-left-symmetric algebra, Hom-Novikov algebras, we will investigate their abelian extensions, central extensions, T*-extensions and deformations.