Extensions of Lie-Rinehart algebras and cotangent bundle reduction

Details Extensions of Lie-Rinehart algebras and cotangent bundle reduction Extensions of Lie-Rinehart algebras and cotangent bundle reduction

2008-01-26

arxiv.org/abs/0801.4099v1

Mathematics

Symplectic Geometry

20 pages

Scientific paper

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 of the cotangent bundle of Q and hence on the ordinary symplectic Poisson algebra of smooth functions on T, and the Poisson algebra of G-invariant functions on T yields a Poisson structure on the space T/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.

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