例外型Yangian代数的R-矩阵实现及其顶点表示

R-矩阵实现是Yangian代数的一种主要实现，它在研究Yangian代数的代数结构和表示理论中有广泛的应用。本项目中我们主要研究例外型Yangian代数的R-矩阵实现及其顶点表示。首先，我们将利用生成元矩阵的高斯分解构造例外型Yangian代数的R-矩阵实现和Drinfeld实现之间的同构映射。其次我们将使用R-矩阵实现来定义例外型Yangian double代数的一个中心扩张，并且以玻色子的形式构造其顶点算子表示。最后，我们将利用R-矩阵的融合方法给出例外型Yangian double代数的中心元的具体表达式。我们还将研究中心元在Harish-Chandra映射下的像以及中心元和表示理论之间的关系。

R-matrix realization is one of the main presentations of the Yangians, which has been widely used to study their algebraic structure and representation theory. In this project, we mainly study the R-matrix realization of the Yangians of exceptional types and their vertex representations. First of all, we will construct the isomorphism between R-matrix realization and Drinfeld realization for the Yangians of exceptional types by applying the Gauss decomposition to the generator matrix. Secondly, we will use the R-matrix realization to define a central extension of the Yangian doubles of exceptional types, and construct their vertex operator representations in terms of bosons. Moreover, central elements of the Yangian doubles of exceptional types will be given explicitly by using the fusion procedure of the R-matrix. We will also investigate the Harish-Chandra images of the central elements as well as the relations between central elements and representation theory.