Cluster X-varieties for dual Poisson-Lie groups I

Details Cluster X-varieties for dual Poisson-Lie groups I Cluster X-varieties for dual Poisson-Lie groups I

2010-05-28

arxiv.org/abs/1005.5289v1

Mathematics

Representation Theory

Scientific paper

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.

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