Fredholm modules for quantum Euclidean spheres

Abstract

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 ).

Characterization of SU q (ℓ + 1)-equivariant spectral triples for the odd dimensional quantum spheres

“…One should mention in this context that the construction by Hawkins & Landi ( [14]) does not deal with equivariance, and more importantly, they produce a (bounded) Fredholm module, not a spectral triple, which is essential for determining the smooth structure, giving a metric on the state space and also help in computing the index map through a local Chern character.…”

The Gysin sequence for quantum lens spaces

“…A set of generators for the K-theory and K-homology of the sphere algebras C(S 2n+1 q ) can be found in [15].…”

Quantum Weighted Projective and Lens Spaces

“…We have denoted by q the deformation parameter, and assume that 0 < q < 1. The original notation of [18] is obtained by setting q = e h/2 ; the generators x i used in [13] are related to ours by x i = z * n+1−i and replacing q → q −1 .…”

“…Irreducible representation of quantum spheres were constructed in [13]. From these, by restriction one gets irreducible representations of quantum lens and weighted projective spaces.…”

“…This is the representation ψ (2n+1) 1 of [13], modulo a renaming of the generators and a redefinition of the parameters.…”