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

Abstract

The odd-dimensional quantum sphere $S_q^{2\ell +1}$ is a homogeneous space for the quantum group ${\mathrm{SU}}_q(\ell +1)$. A generic equivariant spectral triple for $S_q^{2\ell +1}$ 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 ${\mathrm{SU}}_q(2)$, we first study another spectral triple for $S_q^{2\ell +1}$ equivariant under torus group action and constructed by Chakraborty and Pal in [3]. We then derive the results for the ${\mathrm{SU}}_q(\ell +1)$-equivariant triple in the case $q=0$ from those for the torus equivariant triple. For the case $q\ne 0$, we deduce regularity and dimension spectrum from the case $q=0$.