  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.
