Cyclic Cohomology, Quantum group Symmetries and the Local Index Formula for SUq(2)

Abstract

We analyse the noncommutative space underlying the quantum group SUq(2) from the spectral point of view which is the basis of noncommutative geometry, 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. The cochain whose coboundary is the difference between the original Chern character and the local one is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. This specific example allows to illustrate the general notion of locality in NCG. The formulas computing the residue are "local". Locality by stripping all the expressions from irrelevant details makes them computable. The key feature of this spectral triple is its equivariance, i.e. the $SU_q(2)$-symmetry. We shall explain how this leads naturally to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries.