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

TL;DR

This paper studies morphisms of double (quasi-)Poisson algebras, showing their dependence on quivers' undirected structure, and applies these results to describe action-angle duality in classical integrable systems.

Contribution

It establishes a link between morphisms of double (quasi-)Poisson algebras and $H_0$-Poisson structures, and characterizes the algebra structure dependence on quivers.

Findings

Double (quasi-)Poisson algebra structure depends only on the undirected graph of the quiver.

Provides a representation theoretic description of action-angle duality.

Shows that the algebra structure is invariant under certain isomorphisms.

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.