# Some counterexamples in the theory of quantum isometry groups

Research paper by Jyotishman Bhowmick, Debashish Goswami

Indexed on: 25 Oct '09Published on: 25 Oct '09Published in: Mathematics - Operator Algebras

#### Abstract

By considering spectral triples on $S^{2}_{\mu, c}$ ($c>0$) constructed by Chakraborty and Pal (\cite{chak_pal}), we show that in general the quantum group of volume and orientation preserving isometries (in the sense of \cite{goswami2}) for a spectral triple of compact type may not have a $C^*$-action, and moreover, it can fail to be a matrix quantum group. It is also proved that the category with objects consisting of those volume and orientation preserving quantum isometries which induce $C^*$-action on the $C^*$ algebra underlying the given spectral triple, may not have a universal object.