Extensions of Lie-Rinehart algebras and cotangent bundle reduction

Abstract

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 * 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.

Hochschild products and global non-abelian cohomology for algebras. Applications

“…Introduced at the level of groups by Hölder [28], the extension problem is a famous and still open problem to which a vast literature was devoted (see [1] and the references therein). Fundamental results obtained for groups [1,15,37] served as a model for studying the extension problem for several other fields such as Lie/Leibniz algebras [14,31], super Lie algebras [6], associative algebras [17,26], Hopf algebras [8], Poisson algebras [24], Lie-Rinehart algebras [12,25] etc. The extension problem is one of the main tools for classifying 'finite objects' and has been a source of inspiration for developing cohomology theories in all fields mentioned above.…”