标题： 关于杨代数的唯一性 显示英文标题

标题： On the uniqueness of Yangians

摘要： 设$\mathfrak{g}$为复数上的单李代数，令$\mathfrak{g}[u]$表示其多项式流形代数。在20世纪80年代中期，Drinfeld 将$\mathfrak{g}$的杨振宁代数作为在$\mathfrak{g}[u]$上一个自然李双代数结构的量子化问题的唯一解引入。更准确地说，[Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060-1064] 中的定理2断言$\mathfrak{g}[u]$允许唯一的齐次量子化，即$\mathfrak{g}$的杨振宁代数，该代数通过生成元和关系从$\mathfrak{g}$及其伴随表示开始明确描述。尽管杨振宁代数的表示理论此后有了显著发展，但 Drinfeld 定理的完整证明尚未出现。 在本文中，我们提供了对断言的证明，即$\mathfrak{g}[u]$最多接受一个齐次量化。 我们的论证结合了上同调和计算方法，并使用德林费尔德的生成元和一组约简的定义关系来输出任何此类量化的表示。 显示英文摘要

摘要： Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers, and let $\mathfrak{g}[u]$ denote its polynomial current algebra. In the mid-1980s, Drinfeld introduced the Yangian of $\mathfrak{g}$ as the unique solution to a quantization problem for a natural Lie bialgebra structure on $\mathfrak{g}[u]$. More precisely, Theorem 2 of [Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060-1064] asserts that $\mathfrak{g}[u]$ admits a unique homogeneous quantization, the Yangian of $\mathfrak{g}$, which is described explicitly via generators and relations, starting from a copy of $\mathfrak{g}$ and its adjoint representation. Although the representation theory of Yangians has since undergone substantial development, a complete proof of Drinfeld's theorem has not appeared. In this article, we present a proof of the assertion that $\mathfrak{g}[u]$ admits at most one homogeneous quantization. Our argument combines cohomological and computational methods, and outputs a presentation of any such quantization using Drinfeld's generators and a reduced set of defining relations.