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Duality for Lie-Rinehart algebras and the modular class. (English) Zbl 1034.53083
Summary: A notion of homological duality for Lie-Rinehart algebras is studied which generalizes the ordinary duality in Lie algebra (co)homology and in the (co)homology of smooth manifolds. The duality isomorphisms can be given by cap products with suitable fundamental classes and hence may be taken to be natural in any reasonable sense. A precise notion of Poincaré duality, meaning that certain bilinear pairings over appropriate ground rings are nondegenerate, is then introduced, and various examples of Lie-Rinehart algebras are shown to satisfy Poinearé duality. Thereafter a certain intrinsic module introduced by S. Evens, J.-H. Lu and A. Weinstein [Q. J. Math., Oxf. (2) 50, 417–436 (1999; Zbl 0968.58014)] for Lie algebroids is generalized to Lie-Rinehart algebras satisfying duality. This module determines a characteristic class, called the modular class of the Lie-Rinehart algebra; this class lies in an appropriately defined Picard group generalizing the abelian group of flat line bundles on a smooth manifold. A Poisson algebra having the requisite regularity properties determines a corresponding module for its Lie-Rinehart algebra and hence modular class whose square yields the module and characteristic class for its Lie-Rinehart algebra mentioned before. These concepts arise from abstraction from the notion of modular vector field for a smooth Poisson manifold. Finally, it is shown that the Poisson cohomology of certain Poisson algebras satisfies Poincaré duality.

MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
17B63 Poisson algebras
58H05 Pseudogroups and differentiable groupoids
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