Algebraic Geometry and Number Theory

Volume 253 of the series Progress in Mathematics pp 27-68

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

  • V. V. FockAffiliated withInstitute for Theoretical and Experimental Physics
  • , A. B. GoncharovAffiliated withDepartment of Mathematics, Brown University

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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.