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

TL;DR

This paper introduces a natural representation of Lie algebroids on a line bundle, defines the modular class as a cohomology class, and establishes a pairing in Lie algebroid cohomology, extending concepts like orientation and Poincaré duality.

Contribution

It constructs a canonical line bundle representation for Lie algebroids, defines the modular class, and generalizes Poincaré duality pairing in Lie algebroid cohomology.

Findings

Existence of a natural representation on a line bundle Q_A

Definition of the modular class in H^1(A)

A pairing between cohomology spaces generalizing Poincaré duality

Abstract

We show that every Lie algebroid $A$ over a manifold $P$ has a natural representation on the line bundle $Q_A = \wedge^{top}A \otimes \wedge^{top} T^*P$. 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 $Q_A$.