Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems
Annales Henri Lebesgue, Volume 5 (2022), pp. 179-262.


KeywordsDouble bracket, Hamiltonian algebra, Quasi-Hamiltonian algebra, Non-commutative geometry, Integrable system


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.


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