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 $R(\lambda,\mu)T^{1}(\lambda)T^{2}(\mu)=T^{2}(\mu)T^{1}(\lambda)R(\lambda,\mu)$ with $(n_1\times n_2)$-matrices $T(\lambda)$ corresponding to a planar directed network on an annulus. Upon satisfaction of some invertibility conditions, we can extend this construction to realizations of a quantum loop algebra. Having the quantum loop algebra we can also construct a realization of the twisted Yangian algebra, or that of the quantum reflection equation. Every such planar network therefore corresponds to a symplectic leaf of the corresponding infinite-dimensional algebra.