Cluster χ-varieties, amalgamation, and Poisson—Lie groups

  • V. V. Fock
  • A. B. Goncharov
Part of the Progress in Mathematics book series (PM, volume 253)


In this paper, starting from a split semisimple real Lie group G with trivial center, we define a family of varieties with additional structures. We describe them as cluster χ-varieties, as defined in [FG2]. In particular they are Poisson varieties. We define canonical Poisson maps of these varieties to the group G equipped with the standard Poisson—Lie structure defined by V. Drinfeld in [D, D1]. One of them maps to the group birationally and thus provides G with canonical rational coordinates.


Modulus Space Span Tree Simple Root Poisson Structure Braid Group 


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Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • V. V. Fock
    • 1
  • A. B. Goncharov
    • 2
  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Department of MathematicsBrown UniversityProvidenceUSA

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