Transverse measures, the modular class, and a cohomology pairing for Lie algebroids

We show that every Lie algebroid A over a manifold P has a natural representation on the line bundle QA = ∧topA⊗ ∧topT ∗P . The line bundle QA may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of QA may be viewed as transverse measures to A. As a consequence, there is a well-defined class in the first Lie algebroid cohomology H(A) called the modular class of the Lie algebroid A. This is the same as the one introduced earlier by Weinstein using the Poisson structure on A∗. We show that there is a natural pairing between the Lie algebroid cohomology spaces of A with trivial coefficients and with coefficients in QA. This generalizes the pairing used in the Poincare duality of finite-dimensional Lie algebra cohomology. The case of holomorphic Lie algebroids is also discussed, where the existence of the modular class is connected with the Chern class of the line bundle QA.