Extensions of Lie-Rinehart algebras and cotangent bundle reduction

Huebschmann, Johannes
Perlmutter, Matthew
Ratiu, Tudor S.
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Let Q denote a smooth manifold acted upon smoothly by a Lie group G. The G-action lifts to an action on the total space T of the cotangent bundle of Q and hence on the standard symplectic Poisson algebra of smooth functions on T. The Poisson algebra of G-invariant functions on T yields a Poisson structure on the space T/G of G-orbits. We relate this Poisson algebra with extensions of Lie-Rinehart algebras and derive an explicit formula for this Poisson structure in terms of differentials. We then show, for the particular case where the G-action on Q is principal, how an explicit description of the Poisson algebra derived in the literature by an ad hoc construction is essentially a special case of the formula for the corresponding extension of Lie-Rinehart algebras. By means of various examples, we also show that this kind of description breaks down when the G-action does not define a principal bundle.
Comment: The original version has been reworked and expanded with coauthors. The new version has 30 pages; it will appear in the Proceedings of the London Mathematical Society
Mathematics - Symplectic Geometry, Mathematics - Differential Geometry, 53D20, 17B63, 17B65, 17B66, 17B81, 22E70, 53D17, 81S10