标题： 未扭曲量子仿射代数和杨氏代数表示的交织算子的重新研究 显示英文标题

标题： Intertwiners of representations of untwisted quantum affine algebras and Yangians revisited

摘要： 我们讨论$q$-字符在计算$R$-矩阵中的应用。 特别地，我们描述了$R$-矩阵在 E$_8$的第一基本表示的张量平方中的作用，以及在其他一些情况下，张量平方相对于非仿射量子代数的分解具有非平凡重数。 作为说明，我们还恢复了在所有其他类型未扭曲量子仿射代数的第一基本表示的张量平方上无重数情况下的$R$-矩阵。 答案是用与相对于非仿射量子代数的张量平方分解相关的投影算子来表示的。 然后我们给出了$R$-矩阵的显式表达式，这些表达式是相对于自然基的矩阵单位（除了 E$_8$的情况）。 我们给出杨氏代数$R$-矩阵的类似公式。 显示英文摘要

摘要： We discuss applications of the $q$-characters to the computation of the $R$-matrices. In particular, we describe the $R$-matrix acting in the tensor square of the first fundamental representation of E$_8$ and in a number of other cases, where the decomposition of the tensor squares with respect to non-affine quantum algebra has non-trivial multiplicities. As an illustration, we also recover $R$-matrices acting in the multiplicity free-case on the tensor squares of the first fundamental representations of all other types of untwisted quantum affine algebras. The answer is written in terms of projectors related to the decomposition of the tensor squares with respect to non-affine quantum algebras. Then we give explicit expressions for the $R$-matrices in terms of matrix units with respect to a natural basis (except for the case of E$_8$). We give similar formulas for the Yangian $R$-matrices.