"Primary and flat secondary characteristic classes for Lie algebroids, review and problems "

Journal Travaux Math�matiques, University of Luxembourg, 2005 (in print)
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Characteristic homomorphisms of principal fibre bundles (primary and secondary) are invariants of Lie algebroids of principal fibre bundles and can be constructed for all regular Lie algebroids. In this short note we give a survey of the problems for primary characteristic classes of Lie algebroids and a part of problems for secondary characteristic classes which concern "flat" classes. The paper stresses the fact the right approach to the theory of algebroid characteristic classes is not an adaptation of the classical theory of invariant polynomials of a given Lie group and of their representations and then enumeration of the primary and secondary characteristic classes but the study of bundles of polynomials and their invariant cross-sections. We also present some open problems.