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

Abstract

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.

arXiv commentv2: 51 pages, 3 figures -- Sections 2,3,4 rearranged + intro extended + minor changes. Accepted in Ann. Henri Lebesgue

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Cited by in corpus (5)

• Around Van den Bergh's double brackets for different bimodule structures
• Integrable systems on multiplicative quiver varieties from cyclic quivers
• On the noncommutative Poisson geometry of certain wild character varieties
• Functorial constructions related to double Poisson vertex algebras
• Euler continuants in noncommutative quasi-Poisson geometry