Quantum Isometry Groups: Examples and Computations

In this follow-up of the article: Quantum Group of Isometries in Classical and Noncommutative Geometry(arXiv:0704.0041) by Goswami, where quantum isometry group of a noncommutative manifold has been defined, we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori. It is also proved that the quantum isometry group of an isospectral deformation of a (classical or noncommutative) manifold is a suitable deformation of the quantum isometry group of the original (undeformed) manifold.Comment: minor corrections and notational changes made; results of section 3 strengthened by relaxing the assumption of nuclearit

Quantum Isometry groups of the Podles Spheres

For $\mu \in (0,1), c> 0,$ we identify the quantum group $SO_\mu(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries' in the sense of \cite{goswami2} on the natural spectral triple on the Podles sphere $S^2_{\mu, c}$ constructed by Dabrowski, D'Andrea, Landi and Wagner in \cite{{Dabrowski_et_al}}.Comment: Some explanations added in subsections 3.3 and 3.4, to appear in J.Funct.Ana

Compact quantum metric spaces from quantum groups of rapid decay

We present a modified version of the definition of property RD for discrete quantum groups given by Vergnioux in order to accommodate examples of non-unimodular quantum groups. Moreover we extend the construction of spectral triples associated to discrete groups with length functions, originally due to Connes, to the setting of quantum groups. For quantum groups of rapid decay we study the resulting spectral triples from the point of view of compact quantum metric spaces in the sense of Rieffel.Comment: 19 page