  Double (quasi-)Poisson algebras were introduced by Van den Bergh as
non-commutative analogues of algebras endowed with a (quasi-)Poisson bracket.
In this work, we provide a study of morphisms of double (quasi-)Poisson
algebras, which we relate to the $H_0$-Poisson structures of Crawley-Boevey. We
prove in particular that the double (quasi-)Poisson algebra structure defined
by Van den Bergh for an arbitrary quiver only depends upon the quiver seen as
an undirected graph, up to isomorphism. We derive from our results a
representation theoretic description of action-angle duality for several
classical integrable systems.
