Inspired by [3] we introduce the concept of extended Hopf algebra and consider their cyclic cohomology in the spirit of Connes-Moscovici [3, 4, 5]. Extended Hopf algebras are closely related, but different from, Hopf algebroids. Their definition is motivated by attempting to define cyclic cohomology of Hopf algebroids in general. Many of Hopf algebra like structures, including the Connes-Moscov...

Graph cocycles for star-products are investigated from the combinatorial point of view, using Connes-Kreimer renormalization techniques. The Hochschild complex, controlling the deformation theory of associative algebras, is the “Kontsevich representation” of a DGLA of graphs coming from a pre-Lie algebra structure defined by graph insertions (Gerstenhaber composition with Leibniz rule). Propert...

Recent work in perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms defined by the author and W. Zhao, we de...

Let X be a manifold on which a discrete (pseudo)group of diffeomorphisms Γ acts, and let E be a Γ-equivariant vector bundle on X. We give a construction of cyclic cocycles on the cross product algebra C∞ 0 (X)o Γ representing the equivariant characteristic classes of E. Our formulas can be viewed as a higher-dimensional analogue of Connes’ Godbillon-Vey cyclic cocycle. An essential tool for our...

This is a detailed survey on the QWEP conjecture and Connes’ embedding problem. Most of contents are taken from Kirchberg’s paper [Invent. Math. 112 (1993)].

In this paper, we verify the $L^p$ coarse Baum–Connes conjecture for spaces with finite asymptotic dimension $p\in\[1,\infty)$. We also show that $K$-theory of Roe algebras is independent $p\in(1,\infty)$ dimension.