Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems

TL;DR

This paper explores the dual of semisimple Poisson-Lie groups through cluster algebra structures, establishing bases with positive coefficients and linking moduli spaces of G-local systems to cluster Poisson algebras.

Contribution

It demonstrates that the coordinate ring of the dual Poisson-Lie group embeds into a cluster Poisson algebra and identifies the moduli space of G-local systems with a cluster algebra structure.

Findings

Coordinate ring of G* embeds into a cluster Poisson algebra

Existence of a natural basis with positive integer structure coefficients

Moduli space of G-local systems coincides with a cluster Poisson algebra

Abstract

We study the dual ${\rm G}^\ast$ of a standard semisimple Poisson-Lie group ${\rm G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}({\rm G}^\ast)$ can be naturally embedded into a cluster Poisson algebra with a Weyl group action. We prove that $\mathcal{O}({\rm G}^\ast)$ admits a natural basis which has positive integer structure coefficients and satisfies an invariance property with respect to a braid group action. We continue the study of the moduli space $\mathscr{P}_{{\rm G},\mathbb{S}}$ of ${\rm G}$-local systems introduced in \cite{GS3}, and prove that the coordinate ring of $\mathscr{P}_{{\rm G}, \mathbb{S}}$ coincides with its underlying cluster Poisson algebra.