Transverse measures, the modular class and a cohomology pairing for Lie algebroids. (English) Zbl 0968.58014

Evens, Sam; Lu, Jiang-Hua; Weinstein, Alan

Q. J. Math., Oxf. II. Ser. 50, No. 200, 417-436 (1999).

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.

Reviewer: Maido Rahula (Tartu)

MSC:

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)

Keywords:

line bundle; Lie algebroid cohomology

Citations:

Zbl 0810.53019; Zbl 0902.58013