Some Exotic Characteristic Homomorphism for Lie Algebroids

TL;DR

This paper introduces a new secondary characteristic homomorphism for Lie algebroids that generalizes several known exotic characteristic classes, unifying various cases including flat regular Lie algebroids and principal bundles.

Contribution

It defines a universal characteristic homomorphism for pairs of Lie algebroids and flat connections, extending existing theories to more general and arbitrary Lie algebroids.

Findings

Generalizes known exotic characteristic classes for flat Lie algebroids

Introduces a universal characteristic homomorphism for Lie algebroid pairs

Connects various existing characteristic class theories under a unified framework

Abstract

The authors define some secondary characteristic homomorphism for the triple (A,B,\bigtriangledown), in which B\subset A is a pair of regular Lie algebroids over the same foliated manifold and \bigtriangledown:L\rightarrow A is a homomorphism of Lie algebroids (i.e. a flat L-connection in A) where L is an arbitrary (not necessarily regular) Lie algebroid and show that characteristic classes from its image generalize known exotic characteristic classes for flat regular Lie algebroids (Kubarski) and flat principal fibre bundles with a reduction (Kamber, Tondeur). The generalization includes also the one given by Crainic for representations of Lie algebroids on vector bundles. For a pair of regular Lie algebroids B \subset A and for the special case of the flat connection id:A\rightarrow A we obtain a characteristic homomorphism which is universal in the sense that it is a factor of any other one for an arbitrary flat L-connection \bigtriangledown:L\rightarrow A.