任意维度中R矩阵的迭代构造
Iterative construction of the R-matrices in arbitrary dimensions
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中文总结 AI 辅助
本文提出一种特殊ansatz迭代求解常数Yang-Baxter方程，构造四个R矩阵序列，并分析其谱与量子群表示及链环不变量的联系。
中文摘要 AI 辅助
我们研究了一种特殊的ansatz，它允许对常数Yang-Baxter方程进行迭代求解。通过测试这一ansatz，我们构造了四个常数R矩阵序列。在每个序列中，R矩阵作用于维度线性增长的向量空间的张量平方上。每个R矩阵还依赖于一个单一的复参数。通过分析R矩阵的谱，我们得出结论：第一和第三序列分别与量子群U_q(sl(2))和U_q(sl(2|1))的对称张量表示相关。另外两个序列似乎与q为单位根情形下量子群的表示有关。第二序列的元素与U_q(sl(2))的幂零表示相关。我们还检查了这些R矩阵与链环不变量之间的对应关系。将Reshetikhin-Turaev程序应用于第一、第二和第四序列，分别得到彩色Jones多项式、ADO不变量和Alexander多项式。令我们惊讶的是，对第三序列进行类似计算得到的结果是平凡的，恒等于1。
英文摘要
We investigate a special ansats that allows for an iterative solution of the constant Yang-Baxter equation. Testing this ansatz, we construct four sequences of the constant R-matrices. In each sequence the R-matrices act on the tensor squares of vector spaces of linearly growing dimensions. Each R-matrix also depends on a single complex parameter. By analyzing the spectra of the R-matrices, we conclude that the first and the third series are associated with the symmetric tensor representations of the quantum groups U_q(sl(2)) and U_q(sl(2|1)), respectively. The other two series appear to be related to representations of quantum groups in the case where q is a root of unity. The elements of the second series are associated with nilpotent representations of U_q(sl(2)). We also check the correspondence between these R-matrices and link invariants. Applying the Reshetikhin-Turaev procedure to the first, second, and fourth sequences yields, respectively, colored Jones polynomials, ADO invariants, and Alexander polynomials. To our surprise, a similar calculation for the third series yields a trivial result, identically equal to 1.
发表机构
- National Research University ”Higher School of Economics”（国立高等经济大学）
- Bogoliubov Laboratory of Theoretical Physics, JINR（约飞联合核子研究所博戈留博夫理论物理实验室）
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