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

Abstract

The odd-dimensional quantum sphere Sq2ℓ + 1 is a homogeneous space for the quantum group SUq(ℓ + 1). A generic equivariant spectral triple for Sq2ℓ + 1 on its L2-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 SUq(2), we first study another spectral triple for Sq2ℓ + 1 equivariant under torus group action and constructed by Chakraborty and Pal in [3]. We then derive the results for the SUq(ℓ + 1)-equivariant triple in the case q = 0 from those for the torus equivariant triple. For the case q ≠ 0, we deduce regularity and dimension spectrum from the case q = 0.