Cluster variables for affine Lie--Poisson systems

TL;DR

This paper constructs cluster algebra realizations of affine Lie--Poisson systems using planar directed networks, extending to quantum loop algebras and twisted Yangians, linking network topology to algebraic structures.

Contribution

It introduces a novel method to realize affine Lie--Poisson and quantum loop algebras via planar networks on a disc and annulus, connecting network configurations to algebraic symplectic leaves.

Findings

Constructs cluster algebra realizations from planar networks on a disc.

Extends the construction to quantum loop algebras under invertibility conditions.

Links network topology to symplectic leaves of infinite-dimensional algebras.

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.