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

Yang-Baxter type models are integrable deformations of integrable field theories, such as the principal chiral model on a Lie group $G$ or $\sigma$-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 $\mathscr U_q(\mathfrak g)$. 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-Tian-Shansky 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 $\mathscr U_q(\mathfrak g)$, 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.