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

Abstract

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 . 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-Tian-Shansky bracket on the dual group , 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 , including the q-Poisson–Serre relations. We consider reality conditions leading to q being either real or a phase. We determine the non-abelian 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.