2002/09/12 by Alain Connes, Connes, Alain · 2 citations
Mathematics · Physics and Astronomy · #19K33 #46L #58B34 #81R50 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #math-ph #math.MP #math.OA #math.QA #msc:19K33 #msc:46L #msc:58B34 #msc:81R50
paper · pdf · doi:10.48550/arxiv.math/0209142
57 pages
arxiv created 2002/09/12 · openalex publication_date 2002/09/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 SUq(2)-symmetry. We shall explain how this leads naturally to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries.