"New question in the flat secondary characteristic classes, Lie algebroids approach "

Classical characteristic classes on principal bundles (primary and secondary) have Lie algebroid's nature. Firstly it was noticed in 1972 by N.Teleman   for primary classes (under assumption of the connectedness of the structure Lie group). Next, it was continued by the second author since eighties of the previous century (primary and secondary classes) and lastly by R.L.Fernandes and M.Crainic. There are three theories of secondary characteristic classes on Lie algebroids, the first was given by J.Kubarski, the second by R.L.Fernandes   and the third by M.Crainic. The aim of this thak  is to give -- in a spirit of Kamber and Tondeur conception   -- a unification of the theories of "flat characteristic classes" given by J.Kubarski for regular Lie algebroids   with those given by M.Crainic for representations of arbitrary (also irregular) Lie algebroids on vector bundles. We construct the characteristic homomorphism for the triple (A,B,Ñ), where A,B, BÌA, are regular Lie algebroids over the same foliated manifold and Ñ:L→A is a flat L-connection in A with L being an arbitrary (also irregular) Lie algebroid. Taking suitable triples the cases considered earlier by Kamber & Tondeur, Kubarski and Crainic are obtained. For L=A and Ñ=id_A we obtain a new important universal characteristic homomorphism for a pair (A,B), BÌA. The universality means that it factorizes the characteristic homomorphisms for all flat connections. Therefore the fundamental meaning has the question about the injectivity of this universal homomorphism.