In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space $\mathfrak{a}$ satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on $\mathfrak{a}$, generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations.

The recipe involves folding and contracting $\mathfrak{a}$-invariant and $\theta$-twisted symmetric classical dynamical $r$-matrices along an involutive automorphism $\theta$. In case of the universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ we determine the subclass of Schiffmann’s classical dynamical $r$-matrices which are $\mathfrak{a}$-invariant and $\theta$-twisted.

The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.

在本文中，我们给出了如何在复向量空间上构造代数值亚纯函数的三元组的具体方法 $\mathfrak{一种}$满足三个耦合的经典动力学杨-巴克斯特方程和一个相关的经典动力学反射方程。这样的三元组提供了一阶微分算子的一致系统的局部因素$\mathfrak{一种}$，概括渐近边界 Knizhnik-Zamolodchikov-Bernard (KZB) 方程。

配方包括折叠和收缩 $\mathfrak{一种}$- 不变和 $\theta$-扭曲对称经典动力学 $r$-沿对合自同构的矩阵 $\theta$. 在简单李代数的泛包络代数的情况下$\mathfrak{G}$ 我们确定希夫曼经典动力学的子类 $r$-矩阵是 $\mathfrak{一种}$- 不变和 $\theta$-扭曲。

该论文的开头部分强调了渐近（边界）KZB 方程、半单李群的表示理论和可积量子场论之间的联系。