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

TL;DR

This paper proves regularity and computes the dimension spectrum of a specific equivariant spectral triple on odd-dimensional quantum spheres, extending previous constructions and analyzing their spectral properties.

Contribution

It establishes regularity and determines the dimension spectrum for the equivariant spectral triple on odd-dimensional quantum spheres, including detailed algebraic constructions.

Findings

Proved regularity of the spectral triple.

Computed the simple dimension spectrum.

Extended results from the $q=0$ case to $q eq 0$.

Abstract

The odd dimensional quantum sphere $S_q^{2\ell+1}$ is a homogeneous space for the quantum group $SU_q(\ell+1)$. A generic equivariant spectral triple for $S_q^{2\ell+1}$ on its $L_2$ space was constructed by Chakraborty & Pal. We prove regularity for that spectral triple here. We also compute its dimension spectrum and show that it is simple. We give 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 $S_q^{2\ell+1}$ equivariant under torus group action constructed by Chakraborty & Pal. We then derive the results for the $SU_q(\ell+1)$-equivariant triple in the $q=0$ case from those for the torus equivariant triple. For the $q\neq 0$ case, we deduce regularity and dimension spectrum from the $q=0$ case.