Quantum isometries and group dual subgroups

TL;DR

This paper investigates the embedding of duals of discrete groups into compact quantum groups, developing techniques to compute associated group families, with implications for quantum isometry groups of manifolds.

Contribution

It introduces new methods for computing group dual subgroups within compact quantum groups, extending Bichon's classification and impacting the understanding of quantum isometries.

Findings

Developed techniques for computing the family F of groups associated with G

Extended Bichon's classification to matrix quantum groups

Provided insights into quantum isometry groups of manifolds

Abstract

We study the discrete groups $\Lambda$ whose duals embed into a given compact quantum group, $\hat{\Lambda}\subset G$. In the matrix case $G\subset U_n^+$ the embedding condition is equivalent to having a quotient map $\Gamma_U\to\Lambda$, where $F=\{\Gamma_U|U\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon's classification of group dual subgroups $\hat{\Lambda}\subset S_n^+$. These results are motivated by Goswami's notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian group dual isometries.