TOWARDS THE CHERN - WEIL HOMOMORPHISMIN NON COMMUTATIVE DIFFERENTIAL GEOMETRYAndreas

Towards the Chern-weil Homomorphism in Non Commutative Differential Geometry

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...

Cyclic cohomology of Hopf algebras, and a non-commutative Chern-Weil theory

We give a construction of Connes-Moscovici’s cyclic cohomology for any Hopf algebra equipped with a character. Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the Weil complex arises naturally when looking at Hopf algebra actions and invariant higher traces, to give a non-commutative version of the usua...

Non Commutative Chern - Weil Homomorphism

Non-commutative Chern-weil Theory and the Combinatorics of Wheeling

This work applies the ideas of Alekseev and Meinrenken’s Noncommutative Chern-Weil Theory to describe a completely combinatorial and constructive proof of the Wheeling Theorem. In this theory, the crux of the proof is, essentially, the familiar demonstration that a characteristic class does not depend on the choice of connection made to construct it. To a large extent this work may be viewed as...

Super-connections and non-commutative geometry

We show that Quillen’s formalism for computing the Chern character of the index using superconnections extends to arbitrary operators with functional calculus. We thus remove the condition that the operators have, up to homotopy, a gap in the spectrum. This is proved using differential graded algebras and non-commutative differential forms. Our results also give a new proof of the coincidence o...