Extensions of Lie-Rinehart algebras and cotangent bundle reduction

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.

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

References in corpus (4)

• Lie-Rinehart algebras, descent, and quantization
• On the geometry of reduced cotangent bundles at zero momentum
• On Singular Poisson Sternberg Spaces
• Higher homotopies and Maurer-Cartan algebras: Quasi-Lie-Rinehart, Gerstenhaber, and Batalin-Vilkovisky algebras

Cited by in corpus (2)

• Hochschild products and global non-abelian cohomology for algebras. Applications
• The global extension problem, crossed products and co-flag non-commutative Poisson algebras