We characterize all equivariant odd spectral triples for the quantum SU (2) group acting on its L 2 -space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SU q (2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant spectral triples in the quantum case is then used for classical SU (2), and we prove that for p < 4, there does not exist any equivariant spectral triple with nontrivial K-homology class and dimension p acting on the L 2 -space.

The quantum group SU q (ℓ + 1) has a canonical action on the odd dimensional sphere S

Abstract. The odd-dimensional quantum sphere S 2`C1 q is a homogeneous space for the quantum group SU q .`C 1/. A generic equivariant spectral triple for S 2`C1 q on its L 2 -space was constructed by Chakraborty and Pal in [4]. We prove regularity for that spectral triple here. We also compute its dimension spectrum and show that it is simple. We give a 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 2`C1 q equivariant under torus group action and constructed by Chakraborty and Pal in [3]. We then derive the results for the SU q .`C 1/-equivariant triple in the case q D 0 from those for the torus equivariant triple. For the case q ¤ 0, we deduce regularity and dimension spectrum from the case q D 0. (2010). 58B34, 46L87, 19K33. Mathematics Subject Classification

In this article, we construct spectral triples for the C * -algebra of continuous functions on the quantum SU (2) group and the quantum sphere. There has been various approaches towards building a calculus on quantum spaces, but there seems to be very few instances of computations outlined in chapter 6, [5]. We give detailed computations of the associated Connes-de Rham complex and the space of L 2 -forms.

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