# zbMATH — the first resource for mathematics

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.

##### 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
Full Text: