Cyclic cohomology, noncommutative geometry and quantum group symmetries

We give an introduction to the basic notions of noncommutative geometry including the calculus of infinitesimals with operators, cyclic cohomology and the local index formula. We also explain in details how the infinitesimal calculus based on operators gives a natural home for the infinitesimal line element ds of geometry and leads one to the basic notion of spectral triple, which is the basic paradigm of noncommutative geometry. In order to illustrate these general concepts we then analyse the noncommutative space underlying the quantum group SUq (2) from this spectral point of view, and show how the general theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudo-differential calculus, the Wodzciki-type residue, and the local cyclic cocycle giving the index formula. This specific example allows to illustrate the general notion of locality in noncommutative geometry. The formulas computing the residue are "local". Locality by stripping all the expressions from irrelevant details makes them easily computable. The original Chern character is non-local and the cochain whose coboundary is the difference between the original Chern character and the local one is much harder to compute than the local cochains. It is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. The key feature of this spectral triple is its equivariance, i.e. the SUq(2) -symmetry. We explain how this leads naturally to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries and relate this notion to previous work.