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Transverse measures, the modular class and a cohomology pairing for Lie algebroids. (English) Zbl 0968.58014
For any Lie algebroid \(A\) over a manifold \(P\) [see I. Vaisman, “Lectures on the geometry of Poisson manifolds” (1994; Zbl 0810.53019) and A. Weinstein, J. Geom. Phys. 23, No. 3-4, 379-394 (1997; Zbl 0902.58013)], a representation of \(A\) on the line bundle \(Q_A= \wedge^{\text{top}} A\otimes \wedge^{\text{top}}T^*P\) is constructed. In the case when \(A\) is the sub-bundle of \(TP\) tangent to a foliation \({\mathcal F}\), sections of \(Q_A\) are the transverse measures to \({\mathcal F}\), by analogy with the top exterior power of Bott connection.
Two applications are proposed:
1) Every representation of \(A\) on a line bundle defines a ‘characteristic class’ in the first Lie algebroid cohomology of \(A\) with trivial coefficients. For the representation on \(Q_A\) we get the modular class of \(A\). When \(A\) is the cotangent bundle Lie algebroid \(T^*P\) of a Poisson manifold \(P\) we get the representation of \(A\) on the ‘square root’ \(\wedge^{\text{top}}T^*P\) of \(Q_A\). The corresponding characteristic class of \(A\) is then the modular class of the Poisson structure, and the Poisson homology is isomorphic to the Lie algebroid cohomology of \(A=T^*P\) with coefficients in \(\wedge^{\text{top}} T^*P\).
2) A pairing between the Lie algebroid cohomology spaces of \(A\) with trivial coefficients and with coefficients in \(Q_A\), like the Poincaré duality for Lie algebra cohomology and de Rham cohomology, is established.

58H05 Pseudogroups and differentiable groupoids
58A12 de Rham theory in global analysis
58A30 Vector distributions (subbundles of the tangent bundles)
53D17 Poisson manifolds; Poisson groupoids and algebroids
22A22 Topological groupoids (including differentiable and Lie groupoids)
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