Cluster $\mathcal X$-varieties for dual Poisson–Lie groups. I

We associate a family of cluster $\mathcal X$-varieties with 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 with the Lie algebra $\mathfrak g$ of $G$, vertices being labeled by cluster $\mathcal X$-varieties and edges by new Poisson birational isomorphisms on appropriate seed $\mathcal X$-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 $\mathcal X$-tori, called tropical mutations (because they are obtained by a tropicalization of the mutation formula), associated with an enrichment of the combinatorics on double words of the Weyl group $W$ of $G$.