University of Hertfordshire

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On q-deformed symmetries as Poisson-Lie symmetries and application to Yang-Baxter type models

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  • 1606.01712

    Accepted author manuscript, 852 KB, PDF document

  • Benoit Vicedo
  • Marc Magro
  • Francois Delduc
  • Sylvain Lacroix
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Original languageEnglish
Number of pages36
JournalJournal of Physics A: Mathematical and Theoretical
Publication statusPublished - 20 Sep 2016


Yang–Baxter type models are integrable deformations of integrable field theories, such as the principal chiral model on a Lie group G or σ-models on (semi-)symmetric spaces G/F. The deformation has the effect of breaking the global G-symmetry of the original model, replacing the associated set of conserved charges by ones whose Poisson brackets are those of the q-deformed Poisson–Hopf algebra Uq ( ) g . Working at the
Hamiltonian level, we show how this q-deformed Poisson algebra originates from a Poisson–Lie G-symmetry. The theory of Poisson–Lie groups and their actions on Poisson manifolds, in particular the formalism of the non-abelian moment map, is reviewed. For a coboundary Poisson–Lie group G, this non-abelian moment map must obey the Semenov-TianShansky bracket on the dual group G*, up to terms involving central quantities. When the latter vanish, we develop a general procedure linking this Poisson bracket to the defining relations of the Poisson–Hopf algebra
Uq ( ) g , including the q-Poisson–Serre relations. We consider reality conditions leading to q being either real or a phase. We determine the nonabelian moment map for Yang–Baxter type models. This enables to compute the corresponding action of G on the fields parametrising the phase space of these models.


This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at © 2016 IOP Publishing Ltd.

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