Cluster X-varieties for dual Poisson-Lie groups II

TL;DR

This paper explores the cluster X-varieties associated with dual Poisson-Lie groups, focusing on the combinatorics of Artin group actions induced by specific Poisson automorphisms.

Contribution

It extends previous work by detailing the cluster combinatorics related to Artin group actions on dual Poisson-Lie groups.

Findings

Describes the cluster combinatorics involved in Artin group actions.

Connects cluster structures with Poisson automorphisms on G*.

Provides a framework for understanding dual Poisson-Lie groups via cluster varieties.

Abstract

In the prequel of this paper, we have associated a family of cluster X-varieties to the dual Poisson-Lie group(G*,\pi_*) of (G,\pi_G) when (G,\pi_G) is a complex semi-simple Lie group of adjoint type, given with the standard Poisson structure \pi_G and \pi_* is the "dual" Poisson structure defined by the Semenov-Tian-Shansky Poisson bracket on G. We describe here the cluster combinatorics involved into the Artin group action on G* given by the De-Concini-Kac-Procesi Poisson automorphisms.