Cluster X-varieties for dual Poisson-Lie groups I

TL;DR

This paper constructs a family of cluster X-varieties associated with dual Poisson-Lie groups of complex semi-simple Lie groups, using combinatorial structures like the W-permutohedron and introducing new Poisson birational isomorphisms.

Contribution

It introduces a novel framework linking cluster X-varieties, Poisson-Lie groups, and combinatorics via the W-permutohedron and new isomorphisms called saltation and tropicalmutations.

Findings

Established a correspondence between cluster X-varieties and dual Poisson-Lie groups.

Defined new Poisson birational isomorphisms called saltation and tropicalmutations.

Connected combinatorics of double words in Weyl groups to Poisson geometry.

Abstract

We associate a family of cluster X-varieties to the dual Poisson-Lie group G* of a complex semi-simple Lie group G of adjoint type given with the standard Poisson structure. This family is described by the W-permutohedron associated to the Lie algebra g of G: vertices being labeled by cluster X-varieties and edges by new Poisson birational isomorphisms, on appropriate seed X-tori, called saltation. The underlying combinatorics is based on a factorization of the Fomin-Zelevinsky twist maps into mutations and other new Poisson birational isomorphisms on seed X-tori called tropicalmutations, associated to an enrichment of the combinatorics on double words of the Weyl group W of G.