The Weil algebra and the secondary characteristic homomorphism of regular Lie algebroids
Volume 54 / 2001
Abstract
This work gives a construction of the secondary characteristic homomorphism in the category of regular Lie algebroids generalizing the theory of Kamber-Tondeur for foliated principal bundles equipped with reductions. Part I is a preparation and concerns the concept of the Weil algebra ${\mathcal W% }{{\boldsymbol{g} }}$ of the Lie algebra bundle ${{\boldsymbol{g} }}$ adjoint to a regular Lie algebroid $A$. A fundamental role is played by its subalgebra $({\mathcal W}{{% \boldsymbol{g} }}) _{I^{0}}$ of invariant cross-sections with respect to adjoint representations. In Part II we give a construction of characteristic invariants of partially flat regular Lie algebroids, measuring the incompatibility of two geometric structures: a partially flat connection and a Lie subalgebroid. This generalizes the classical construction of Kamber-Tondeur. Fundamental properties, for example, independence of the homotopy class of a Lie subalgebroid, are given.
A comparison of the presented Lie algebroid theory with characteristic classes of foliated principal bundles shows the algebroid nature of the latter. For globally flat connnections the concept reduces to characteristic invariants of flat regular Lie algebroids in \cite{Kflat}.
In the Appendix we present the elementary theory of regular Lie algebroids in which the key role is played by a global theorem on solutions of some system of partial differential equations with parameters. One of the main structure theorems concerns invariant cross-sections on ${\mathbb R}\times M$, the basic fact needed in the proof of homotopy independence of characteristic classes.
