Cluster variables for affine Lie–Poisson systems

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.