Non-commutative integrable systems on $b$-symplectic manifolds

TL;DR

This paper explores non-commutative integrable systems on $b$-symplectic manifolds, providing examples and proving an action-angle theorem near Liouville tori within the critical set of the Poisson structure.

Contribution

It introduces the study of non-commutative integrable systems on $b$-symplectic manifolds and proves an action-angle theorem in this context, extending classical results.

Findings

Examples of such systems from manifolds with boundary

An action-angle theorem near Liouville tori in the critical set

Extension of integrable systems theory to $b$-symplectic manifolds

Abstract

In this paper we study non-commutative integrable systems on $b$-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering non-commutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and we prove an action-angle theorem for non-commutative integrable systems on a $b$-symplectic manifold in a neighbourhood of a Liouville torus inside the critical set of the Poisson structure associated to the $b$-symplectic structure.