The uniqueness theorem for the universal R-matrix
The uniqueness theorem for the universal R-matrix
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通用 R 矩阵的唯一性定理
DOI：
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
V. Tolstoy
中科院分区：
文献类型：
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作者：
S. Khoroshkin;V. Tolstoy
到目前为止，量子化Kac-Moody代数的通称r -矩阵这里的lsKac-Moody代数
s包括了所有半简单有限维李代数和所有无限维仿射Kac-Moody代数。被认为是由拟协性和拟三角性的性质唯一决定的（对于某些解）。在这里我们证明了泛r矩阵（对于相同的矩阵）是唯一地由拟共交换性的性质决定的。因此，泛r矩阵的拟三角形性质（以及Yang-Baxter方程！）是拟协易性线性方程的结果。基于量子化代数的Verma模的张量积中的奇异向量的性质和极值投影的结构进行了证明。给出了量子化代数Uq（Ainf1sup(1)）和Uq（Ainf2sup(2)）的泛r矩阵的显式表达式。
Up to now, the universal R-matrix for quantized Kac-Moody algebrasHere the name lsKac-Moody algebras
s includes all semisimple finite-dimensional Lie algebras and all infinite-dimensional affine Kac-Moody algebras. is believed to be uniquely determined (for some ansatz) by properties of a quasi-cocommutativity and a quasi-triangularity. We prove here that the universal R-matrix (for the same ansatz) is uniquely determined by the property of the quasi-cocommutativity only. Thus, the quasi-triangular property (and the Yang-Baxter equation!) for the universal R-matrix is a consequence of the linear equation of the quasi-cocommutativity. The proof is based on properties of singular vectors in the tensor product of the Verma modules and the structure of extremal projector for quantized algebras. Explicit expressions of the universal R-matrix for quantized algebras Uq(Ainf1sup(1)) and Uq(Ainf2sup(2)) are given.