"The Weil algebra and the secondary characteristic homomorphism of regular Lie algebroids"

Lie Algebroids and Related Topics in Differential Geometry, Banach Center Publicationes, Volume 54, 135-173, IMPAN Warszawa 2001.
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This work give 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 to this aim and concern a concept of Weil algebra W(g) of the Lie algebra bundle g adjoint to a regular Lie algebroid A. Foundamental role is played by its subalgebra W(g)_I^o of invariant cross-sections with respect to adjoint representations. In Part II of this work 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 given Lie subalgebroid. This generalize the classical construction of Kamber-Tondeur. Fundamental properties, as for example, independence on the homotopic class of a Lie subalgebroid are given.
A comparison of the presentation Lie algebroid theory with characteristic classes of foliated principal bundles shows the algebroid nature of these last. For the globally flat connnections the concept reduces to characteristic invariants of flat regular Lie algebroids.
In the Appendix we present 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 theorem concern invariant cross-sections on R×M – the base fact needed in the proof of homotopic independence of characteristic classes.