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This article is cited in 5 scientific papers (total in 5 papers)
Research Papers
Cluster $\mathcal X$-varieties for dual Poisson–Lie groups. I
R. Brahami Institut Mathématiques de Bourgogne, Dijon, France
Abstract:
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.
Received: 22.09.2009
Citation:
R. Brahami, “Cluster $\mathcal X$-varieties for dual Poisson–Lie groups. I”, Algebra i Analiz, 22:2 (2010), 14–104; St. Petersburg Math. J., 22:2 (2011), 183–250
Linking options:
https://www.mathnet.ru/eng/aa1177 https://www.mathnet.ru/eng/aa/v22/i2/p14
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Abstract page: | 391 | Full-text PDF : | 110 | References: | 63 | First page: | 5 |
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