Cluster variables for affine Lie–Poisson systems

We show that having any planar (cyclic or acyclicm) 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 $R(\lambda,\mu) {\stackrel{1}{T}} (\lambda) {\stackrel{1}{T}}(\mu)= {\stackrel{2}{T}}(\mu) {\stackrel{1}{T}}(\lambda)R(\lambda,\mu)$ 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.