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 Q A = top A top T * P . The line bundle Q A may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of Q A may be viewed as transverse measures to A. As a consequence, there is a well-defined class in the first Lie algebroid cohomology H 1 (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 Q A . 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 Q A .