Torus Equivariant Spectral Triples for Odd-Dimensional Quantum Spheres Coming from C *-Extensions

Abstract

The torus group (S 1 ) ℓ+1 has a canonical action on the odd dimensional sphere S 2ℓ+1 q . We take the natural Hilbert space representation where this action is implemented and characterize all odd spectral triples acting on that space and equivariant with respect to that action. This characterization gives a construction of an optimum family of equivariant spectral triples having nontrivial K-homology class thus generalizing our earlier results for SU q (2). We also relate the triple we construct with the C * -extension

Quantum double suspension and spectral triples

“…Now if we quantum double suspend the spectral triple (C ∞ (T), L 2 (T), 1 i d dθ ) we get the torus equivariant spectral triple on C(S 2 +1 q ) [5]. Now Theorem 5.7 implies that the torus equivariant spectral triple for the odd-dimensional quantum sphere C(S 2 +1 q ) satisfies topological weak heat kernel asymptotic expansion property with dimension + 1.…”

“…In this section we recall the spectral triple for the odd-dimensional quantum spheres given in [5]. We begin with some known facts about odd-dimensional quantum spheres.…”

Spectral Metric Spaces on Extensions of C*-Algebras

“…The noncommutative spheres have garnered a lot of attention in the literature as examples of noncommutative manifolds and many spectral triples have been suggested (e.g. [5], [7], [17]), though most of these from a very different perspective to ours, for example by looking at the representation theory of the ordinary SU(2) group and focusing on those triples which behave equivariantly with respect to the group co-action.…”

“…The named authors show that any spectral triple on C(SU q (2)) which is of a certain natural form and which is equivariant for the quantum group co-action of SU q (2) must have spectral dimension at least 3, which is in contrast to our spectral triple of dimension 2. In [7] the same authors construct spectral triples on C(SU q (2)) using an altogether different approach, focusing on those triples which are equivariant for the action of T 2 on C(SU q (2)), which might be closer to our spectral triple. The construction in [6] was used and further developed by Connes [14].…”

Regularity and dimension spectrum of the equivariant spectral triple for the odd-dimensional quantum spheres

“…We give detailed construction of its smooth function algebra and some related algebras that help proving regularity and in the computation of the dimension spectrum. Following the idea of Connes for SU q (2), we first study another spectral triple for S 2ℓ+1 q equivariant under torus group action constructed by Chakraborty & Pal in [3]. We then derive the results for the SU q (ℓ + 1)-equivariant triple in the q = 0 case from those for the torus equivariant triple.…”