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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

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


Author: R. Brahami
Original publication: Algebra i Analiz, tom 22 (2010), nomer 2.
Journal: St. Petersburg Math. J. 22 (2011), 183-250
MSC (2010): Primary 22E70, 81R10
Published electronically: February 8, 2011
MathSciNet review: 2668124
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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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Additional Information

R. Brahami
Affiliation: Institut Mathématique de Bourgogne, Dijon, France
Email: Renaud.Brahami@u-bourgogne.fr

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01138-0
PII: S 1061-0022(2011)01138-0
Keywords: Cluster combinatorics, Poisson structure, tropical mutation, saltations
Received by editor(s): September 22, 2009
Published electronically: February 8, 2011
Article copyright: © Copyright 2011 American Mathematical Society