Weakly Hamiltonian actions

TL;DR

This paper extends the theory of non-commutative integrable systems to weakly Hamiltonian actions on Poisson manifolds, showing how abelian actions decompose on symplectic manifolds and exploring broader Poisson generalizations.

Contribution

It introduces a novel framework for weakly Hamiltonian actions, demonstrating their splitting into Hamiltonian and non-Hamiltonian components on symplectic manifolds.

Findings

Abelian weakly Hamiltonian actions split into Hamiltonian and non-Hamiltonian parts.

Generalizations of these actions are developed for Poisson manifolds.

Theoretical framework broadening integrable systems in Poisson geometry.

Abstract

In this paper we generalize constructions of non-commutative integrable systems to the context of weakly Hamiltonian actions on Poisson manifolds. In particular we prove that abelian weakly Hamiltonian actions on symplectic manifolds split into Hamiltonian and non-Hamiltonian factors, and explore generalizations in the Poisson setting.