Poisson Structures and Potentials

  • Anton AlekseevEmail author
  • Arkady Berenstein
  • Benjamin Hoffman
  • Yanpeng Li
Part of the Progress in Mathematics book series (PM, volume 326)


We introduce the notion of weakly log-canonical Poisson structures on positive varieties with potentials. Such a Poisson structure is log-canonical up to terms dominated by the potential. To a compatible real form of a weakly logcanonical Poisson variety, we assign an integrable system on the product of a certain real convex polyhedral cone (the tropicalization of the variety) and a compact torus. We apply this theory to the dual Poisson-Lie group G* of a simply-connected semisimple complex Lie group G.

We define a positive structure and potential on G* and show that the natural Poisson-Lie structure on G* is weakly log-canonical with respect to this positive structure and potential. For KG the compact real form, we show that the real form K* ⊂ G* is compatible and prove that the corresponding integrable system is defined on the product of the decorated string cone and the compact torus of dimension \( \frac{1}{2} \) (dimG − rankG).


Poisson structures Poisson-Lie groups Potentials Tropicalization 

Mathematics Subject Classification (2010):

20G42 53D17 


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Anton Alekseev
    • 1
    Email author
  • Arkady Berenstein
    • 2
  • Benjamin Hoffman
    • 3
  • Yanpeng Li
    • 1
  1. 1.Section of MathematicsUniversity of GenevaGen`eve 4Switzerland
  2. 2.Department of MathematicsUniversity of OregonEugeneUSA
  3. 3.Department of MathematicsCornell UniversityIthacaUSA

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