与一个泛二值代数群相关的量子Yang–Baxter方程的解
A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group
- Steklov Mathematical Institute of the Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
- Steklov International Mathematical Center（斯捷克洛夫国际数学中心）
- Centre of Integrable Systems, P.G. Demidov Yaroslavl State University（雅罗斯拉夫尔国立大学普·格·杰米多夫可积系统中心）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该文从泛二值代数群的结合性构造量子Yang–Baxter方程的谱参数解，得到满足WDVV方程的9×9幺正R矩阵，并显式处理A3情形。
中文摘要 AI 辅助
我们从泛对称$2$-代数二值群的结合性条件构造量子Yang–Baxter方程的谱参数解，该群的乘法法则依赖于四个系数$k_2,k_4,k_6,k_8$，并满足单一关系$4k_8-k_4^2+k_2k_6=0$——该关系与Dubrovin归一化势的三维WDVV方程一致。每个具有循环向量场的三维Frobenius流形都承载这样的谱二值群，其有限分支点是乘法算子的特征值。对于任意含单位代数，我们证明由乘法和单位构造的双参数谱$R$-算子的Yang–Baxter缺陷正比于一个由结合子构成的显式张量；对于切Frobenius代数，这产生一个正则的、幺正的$9\ imes9$ $R$-矩阵，在WDVV点精确满足量子Yang–Baxter方程，其谱参数是双椭圆曲线的Abel坐标，该曲线的椭圆商是谱曲线，并伴随一个可积链。$A_3$ Frobenius流形被显式地计算出来。
英文摘要
We construct spectral-parameter solutions of the quantum Yang--Baxter equation from the associativity condition of the universal symmetric $2$-algebraic two-valued group, whose multiplication law depends on four coefficients $k_2,k_4,k_6,k_8$ subject to the single relation $4k_8-k_4^2+k_2k_6=0$ -- which coincides with the three-dimensional WDVV equation for Dubrovin's normalised potential. Every three-dimensional Frobenius manifold with a cyclic vector field carries such a spectral two-valued group, whose finite branch points are the eigenvalues of the multiplication operator. For an arbitrary unital algebra we show that a two-parameter spectral $R$-operator built from multiplication and unit has Yang--Baxter defect proportional to an explicit tensor built from the associator; for tangent Frobenius algebras this yields a regular, unitary $9\times9$ $R$-matrix satisfying the quantum Yang--Baxter equation exactly at WDVV points, with spectral parameter the Abel coordinate of a bielliptic curve whose elliptic quotient is the spectral curve, together with an associated integrable chain. The $A_3$ Frobenius manifold is worked out explicitly.
