Towards the Chern-weil Homomorphism in Non Commutative Differential Geometry
نویسندگان
چکیده
In this short review article we sketch some developments which should ultimately lead to the analogy of the Chern-Weil homomorphism for principle 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. See [9] for an account of the classical theory using this approach. In this paper we give an outline of the construction of a non commutative analogy of the Frölicher–Nijenhuis bracket as well as some simple applications. For simplicity we work in a purely algebraic setting but the whole theory can also be developed for topological algebras as well as for the so called convenient algebras (see [5]) which are best suited for differentiation and take care of completed tensor products. For a detailed exposition in the latter setting see [1] and [2].
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