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

摘要

The odd-dimensional quantum sphere S_q~(2?+1) is a homogeneous space for the quantum group SU_q(?+1). A generic equivariant spectral triple for S_q~(2?+1) 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_q~(2?+1) equivariant under torus group action and constructed by Chakraborty and Pal in [3]. We then derive the results for the SU _q(?+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.