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In this paper we analyze the obstructions to the existence of globalaction-angle variables for regular non-commutative integrable systems (NCIsystems) on Poisson manifolds. In contrast with local action-angle variables,which exist as soon as the fibers of the momentum map of such an integrablesystem are compact, global action-angle variables rarely exist. This fact wasfirst observed and analyzed by Duistermaat in the case of Liouville integrablesystems on symplectic manifolds and later by Dazord-Delzant in the case ofnon-commutative integrable systems on symplectic manifolds. In our more generalcase where phase space is an arbitrary Poisson manifold, there are moreobstructions, as we will show both abstractly and on concrete examples. Ourapproach makes use of a few new features which we introduce: the action bundleand the action lattice bundle of the NCI system (these bundles are canonicallydefined) and three foliations (the action, angle and transverse foliation),whose existence is also subject to obstructions, often of a cohomologicalnature.

Global Action-Angle Variables for Non-Commutative Integrable Systems