Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems

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