The Yang-Baxter equation and invariants of links
The Yang-Baxter equation and invariants of links
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DOI：
10.1007/bf01393746
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发表时间：
1988-10
影响因子：
3.1
通讯作者：
V. Turaev
中科院分区：
文献类型：
--
作者：
V. Turaev
杨-巴克斯特方程最早出现在20世纪60年代末至70年代初的CN Yang和RJ巴克斯特的独立论文中。该方程及其解在完全可积量子系统理论和统计力学精确求解模型理论中起着基础性作用（见[1，9]）。杨-巴克斯特方程和链环的新多项式不变量之间的关系已经隐含在Jones的先驱论文[5]中。在该文件琼斯介绍了他著名的多项式的联系通过研究某些有限维杨诺依曼代数。D的一句话。Evans在[5]中指出，这些代数是物理学家们在研究统计力学的Potts模型和冰型模型时发现的，[5]出现后，几位作者引入了两个新的链P和F的合痕不变量，它们（直到重新参数化）是二元Laurent多项式（见I-3，7]）。P和F都包含Jones多项式，但不能从它推导出来。已知的P的构造要么诉诸于von Neumann代数，要么诉诸于Hecke代数，要么诉诸于基于Conway型关系的几何迭代过程。唯一已知的F的构造，由于考夫曼[7]，呼吁一个类似的几何过程。最近，Jones [6]证明了P可以用Hecke代数的显式矩阵表示来构造，这是在量子力学的著作中引入的。逆散射方法，并与杨-巴克斯特方程。在[6]中强调，”多项式的一致的一般图像开始出现，相关的数学形式是量子逆散射方法和量子统计力学”。
The Yang-Baxter equation first appeared in the independent papers of CN Yang and RJ Baxter in the late 1960's-early 1970's. This equation and its solutions play fundamental role in the theory of completely integrable quantum systems and in the theory of exactly solved models of statistical mechanics (see [1, 9]). A relationship between the Yang-Baxter equation and the new polynomial invariants of links was implicit already in the pioneer paper of Jones [5]. In that paper Jones introduced his famous polynomial of links via a study of certain finite dimensional yon Neumann algebras. A remark of D. Evans mentioned in [5] points out that these algebras were earlier discovered by physicists who used them to study the Potts model and the ice-type model of statistical mechanics.After appearance of [5] several authors introduced two new isotopy invariants of links P and F which are (up to reparametrization) Laurent polynomials of 2 variables (see I-3, 7]). Both P and F contain the Jones polynomial but can not be deduced from it. Known constructions of P appeal either to von Neumann algebras, or to Hecke algebras, or to a geometric iterative procedure based on a Conway-type relation. The only known construction of F, due to Kauffman [7], appeal to an analogous geometric procedure. Recently, Jones [6] has shown that P can be constructed using explicit matrix representations of Hecke algebras, introduced in works on the quantum! nverse scattering method and related to the Yang-Baxter equation. It is stressed in [6] that" a consistent general picture of the polynomials is starting to emerge, the relevant mathematical formalism being quantum inverse scattering method and quantum statistical mechanics".