Abstract
In this short review article we sketch some developments which should ultimately lead to the analogy of the Chern-Weil homomorphism for principal bundles in the realm of non-commutative differential geometry. Principal bundles there should have Hopf algebras as structure `cogroups'. Since the usual machinery of Lie algebras, connection forms, etc\., just is not available in this setting, we base our approach on the Frölicher--Nijenhuis bracket.
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