Fredholm modules for quantum Euclidean spheres

The quantum Euclidean spheres, S-q(N-1), are (noncommutative) homogeneous spaces of quantum orthogonal groups, SOq(N). The *-algebra A(S-q(N-1)) of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, umpotent 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(N-1). We also construct the corresponding Chem characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i.e., for N 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(N-1)) (C) 2003 Elsevier B.V. All rights reserved.