Geodesic
Cluster $\mathcal X$-varieties for dual Poisson--Lie groups.~I
Algebra i analiz, Tome 22 (2010) no. 2, pp. 14-104.

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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 $G$.
Keywords: cluster combinatorics, Poisson structure, tropical mutation, saltations. 
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R. Brahami. Cluster $\mathcal X$-varieties for dual Poisson--Lie groups.~I. Algebra i analiz, Tome 22 (2010) no. 2, pp. 14-104. http://geodesic.mathdoc.fr/item/AA_2010_22_2_a1/

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