Duality for Lie-Rinehart algebras and the modular class

Details Duality for Lie-Rinehart algebras and the modular class Duality for Lie-Rinehart algebras and the modular class

1997-02-08

arxiv.org/abs/dg-ga/9702008v2

J. reine angew. Math. 510 (1999), 103-159

Mathematics

Differential Geometry

55pages, AMSTeX 2.1

Scientific paper

We introduce a notion of duality for a Lie-Rinehart algebra giving certain bilinear pairings in its cohomology generalizing the usual notions of Poincar\'e duality in Lie algebra cohomology and de Rham cohomology. We show that the duality isomorphisms can be given by a cap product with a suitable fundamental class and hence may be taken natural in any reasonable sense. Thereafter, for a Lie-Rinehart algebra satisfying duality, we introduce a certain intrinsic module which provides a crucial ingredient for the construction of the bilinear pairings. This module determines a certain class, called modular class of the Lie-Rinehart algebra, which lies in a certain Picard group generalizing the abelian group of flat line bundles on a smooth manifold. Finally, we show that a Poisson algebra having suitable properties determines a certain module for the corrresponding Lie-Rinehart algebra and hence modular class whose square yields the module and characteristic class for its Lie-Rinehart algebra mentioned before. In particular, this gives rise to certain bilinear pairings in Poisson homology.

Affiliated with

Also associated with

No associations

Rating

Duality for Lie-Rinehart algebras and the modular class does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Duality for Lie-Rinehart algebras and the modular class, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Duality for Lie-Rinehart algebras and the modular class will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-210749