Extensions of Lie-Rinehart algebras and cotangent bundle reduction

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*Q of the cotangent bundle of Q and hence on the standard symplectic Poisson algebra of smooth functions on T*Q. The Poisson algebra of G-invariant functions on T*Q yields a Poisson structure on the space (T*Q)/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.