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

Brahami, R.

doi:10.1090/S1061-0022-2011-01138-0

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

St. Petersburg Math. J. 22 (2011), 183-250

DOI: https://doi.org/10.1090/S1061-0022-2011-01138-0

Published electronically: February 8, 2011

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Abstract:

We associate a family of cluster $\mathcal X$-varieties with the dual Poisson–Lie group $G^*$ of a complex semisimple 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 $G$.

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