Fredholm modules for quantum Euclidean spheres

✍ Scribed by Eli Hawkins; Giovanni Landi

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 186 KB
• Volume: 49
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(03)00092-5

✦ Synopsis

The quantum Euclidean spheres, S N-1 q , are (noncommutative) homogeneous spaces of quantum orthogonal groups, SO q (N). The * -algebra A(S N-1 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 N-1 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., 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 N-1 q