Fredholm Modules for Quantum Euclidean Spheres

The quantum Euclidean spheres, S q , are (noncommutative) homogeneous spaces of quantum orthogonal groups, SOq(N). The ∗-algebra A(S q ) of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres S q . We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i. e., forN even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra A(S q ). 2000 Mathematics Subject Classi£cation: 19D55; Secondary 20G42, 58B34.