"Representations of Lie algebroids and characteristic classes"

Lecture, The Fourth International Conference SYMMETRY IN NONLINEAR MATHEMATICAL PHYSICS, July 9-15, 2001, Kyiv, Ukraine.
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The talk concerns:

• some class of infinite dimensional representations on the algebra of Covariant Differential Operators of a vector bundle E ® M and
• obstructions to their existence (primary characteristic classes); next I give some important examples of representations – generalizations of the Bott connections and in the end
• I formulate a philosophy of the so-called secondary characteristic classes and three approaches to them.