We construct natural representations of the Connes-Kreimer Lie algebras on rooted trees/Feynman graphs arising from Hecke correspondences in the categories LRF ,LFG constructed by K. Kremnizer and the author. We thus obtain the insertion/elimination representations constructed by Connes-Kreimer as well as an isomorphic pair we term top-insertion/top-elimination. We also construct graded finite-...

To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we associate ”noncommutative varieties” called braided spheres. An example of such a variety is the Podles’ nonstandard quantum sphere. On any braided sphere we introduce and compute an ”equivariant” analogue of Connes’ noncommutative index. In contrast with the Connes’ construction our version of equivariant NC index is based ...

The Connes formula giving the dual description for the distance between points of a Riemannian manifold is extended to the Lorentzian case. It resulted that its validity essentially depends on the global structure of spacetime. The duality principle classifying spacetimes is introduced. The algebraic account of the theory is suggested as a framework for quantization along the lines proposed by ...

We give a rigorous proof that the (codimension one) Connes-Moscovici Hopf algebra HCM is isomorphic to a bicrossproduct Hopf algebra linked to a group factorisation of the diffeomorphism group Diff(R). We construct a second bicrossproduct UCM equipped with a nondegenerate dual pairing with HCM. We give a natural quotient Hopf algebra kλ[Heis] of HCM and Hopf subalgebra Uλ(heis) of UCM which aga...

1 2 CONNES AND MARCOLLI

Around 1980 Connes extended the notions of geometry to the noncommutative setting. Since then non-commutative geometry has turned into a very active area of mathematical research. As a first non-trivial example of a noncommutative manifold Connes discussed subalgebras of rotation algebras, the socalled non-commutative tori. In the last two decades researchers have unrevealed the relevance of no...

LetG be a discrete group and letX be aG-finite, properG-CW-complex. We prove that Kasparov’s equivariant K-homology groups KK∗ (C0(X),C) are isomorphic to the geometric equivariant K-homology groups of X that are obtained by making the geometric K-homology theory of Baum and Douglas equivariant in the natural way. This reconciles the original and current formulations of the Baum-Connes conjectu...

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have dis...

In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spinc geometry depending on whether the geometry is “real” or not. We attempt to flesh out the details of Connes’ ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemann...

We study an equivariant co-assembly map that is dual to the usual Baum–Connes assembly map and closely related to coarse geometry, equivariant Kasparov theory, and the existence of dual Dirac morphisms. As applications, we prove the existence of dual Dirac morphisms for groups with suitable compactifications, that is, satisfying the Carlsson–Pedersen condition, and we study a K–theoretic counte...