Extensions of Lie-Rinehart algebras and cotangent bundle reduction

Let Q be a smooth manifold acted upon smoothly by a Lie group G, and let N be the space of G-orbits. The G-action lifts to an action on the total space T∗Q of the cotangent bundle of Q and hence on the ordinary symplectic Poisson algebra of smooth functions on T∗Q, and the Poisson algebra of G-invariant functions on T∗Q yields a Poisson structure on the space (T∗Q) / G of G-orbits. We develop a description of this Poisson structure in terms of the orbit space N and suitable additional data. When the G-action on Q is principal, the problem admits a simple solution in terms of extensions of Lie-Rinehart algebras. In the general case, extensions of Lie-Rinehart algebras do not suffice, and we show how the requisite supplementary information can be recovered from invariant theory. Subject classification: Primary: 53D20; Secondary: 17B63 17B65 17B66 17B81 22E70 53D17 81S10