Title :

Extensions of Lie-Rinehart algebras and cotangent bundle reduction

Author(s) :

Huebschmann, Johannes [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Perlmutter, Matthew [Auteur]
Ratiu, Tudor S. [Auteur]

English keyword(s) :

Lie-Poisson algebra
Lie-Rinehart algebra
extension of Lie-Rinehart algebra
singular reduction
singular cotangent bundle reduction
semi-algebraic set
Poisson structure
tautological Poisson structure
symplectic leaf
fiber bundle
connection

HAL domain(s) :

Mathématiques [math]/Géométrie symplectique [math.SG]
Mathématiques [math]/Géométrie différentielle [math.DG]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

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

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL