"Pontryagin algebra of a transitive Lie algebroid"

Proceedings of the Winter School on Geometry and Physics, SRNI, 7-14 January, 1989, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II - numero 22 - 1989, 117-126.
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The Chern-Weil homomorphism h^P of a principal fibre bundle P has been known for some forty years (S.S.Chern 1950). On the other hand, in analogy to the theory of Lie groups and Lie algebras, each principal fibre bundle P has its algebraic equivalent: a transitive Lie algebroid A(P) - constructed on the basis of the right-invarint vector fields on P. A(P) is simply a vector bundle equipped with some structures like a structure of a Lie algebra in the module of sections. In turns out that the Chern-Weil homomorphism of P is a notion of the Lie algebroid A(P) of P (provided that P is connected), one can uniquely reproduce the ring of invariant polynomials (Vg*)_I and the Chern-Weil homomorphism h^P: (Vg*)_I ® H(M) (g denotes the Lie algebra of G).
We pay our attention to the fact that this holds although in the Lie algebroid A(P) there is no direct information about the structural Lie group of P which can be disconnected. In addition, we point out that:

a transitive Lie algebroid is a simple structure than a principal fibre bundle (nonisomorphic principal fibre bundles can possess isomorphic Lie algebroids),
there exists nonintegrable transitive Lie algebroids, i.e. which cannot be realized as the Lie algebroids of principal fibre bundles.

In connection with the above it is important to construct the Chern-Weil homomorphism of a transitive Lie algebroid. This is the aim of this paper.
In the end of this paper we mention that there exists a characteristic homomorphism of flat (and of partially flat) regular Lie algebroids. The papers [29] and [39], below, are devoted to detailed descriptions of these homomorphisms.