Abstract

We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all \(n_1+m\) sources are separated from all \(n_2+m\) sinks, we can construct a cluster-algebra realization of elements of an affine Lie–Poisson algebra \(R(\lambda,\mu){\stackrel{1}{T}}(\lambda){\stackrel{2}{T}}(\mu) ={\stackrel{2}{T}}(\mu){\stackrel{1}{T}}(\lambda)R(\lambda,\mu)\) with \(({n_1\times n_2})\)-matrices \(T(\lambda)\). Upon satisfaction of some invertibility conditions, we can construct a realization of a quantum loop algebra. Having the quantum loop algebra, we can also construct a realization of the twisted Yangian algebra or of the quantum reflection equation. For each such a planar network, we can therefore construct a symplectic leaf of the corresponding infinite-dimensional algebra.

中文翻译：

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仿射李-泊松系统的聚类变量

摘要

我们证明，在圆盘上拥有任何平面（循环或非循环）有向网络，唯一条件是所有\(n_1+m\)源与所有\(n_2+m\)汇分离，我们可以构造一个簇代数仿射李-泊松代数元素的实现\(R(\lambda,\mu){\stackrel{1}{T}}(\lambda){\stackrel{2}{T}}(\mu) ={ \stackrel{2}{T}}(\mu){\stackrel{1}{T}}(\lambda)R(\lambda,\mu)\)与\(({n_1\times n_2})\) - 矩阵\(T(\lambda)\)。在满足一些可逆性条件后，我们可以构造量子环代数的实现。有了量子环代数，我们还可以构造扭曲杨代数或量子反射方程的实现。因此，对于每个这样的平面网络，我们可以构造相应的无限维代数的辛叶。