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

### Arupkumar Pal

Indian Statistical Institute, New Delhi### S. Sundar

Institute of Mathematical Sciences, Chennai

## Abstract

The odd-dimensional quantum sphere $S_{q}$ is a homogeneous space for the quantum group $SU_{q}(ℓ +1)$. A generic equivariant spectral triple for $S_{q}$ 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 $S_{q}$ equivariant under torus group action and constructed by Chakraborty and Pal in [3]. We then derive the results for the $SU_{q}(ℓ +1)$-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 $q=0$.

## Cite this article

Arupkumar Pal, S. Sundar, Regularity and dimension spectrum of the equivariant spectral triple for the odd-dimensional quantum spheres. J. Noncommut. Geom. 4 (2010), no. 3, pp. 389–439

DOI 10.4171/JNCG/61