Extensions of Lie-Rinehart algebras and cotangent bundle reduction

TL;DR

This paper explores the relationship between Poisson structures on cotangent bundles under group actions and extensions of Lie-Rinehart algebras, providing explicit formulas and analyzing special cases like principal G-actions.

Contribution

It establishes a connection between Poisson algebras of invariant functions and Lie-Rinehart algebra extensions, offering explicit formulas and clarifying their applicability.

Findings

Explicit formula for Poisson structure in terms of differentials.

The formula applies to principal G-actions but not to non-principal actions.

Examples illustrating the breakdown of the description outside principal cases.

Abstract

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.