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

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