Quantum isometries and group dual subgroups

Banica, Teodor; Bhowmick, Jyotishman; De Commer, Kenny

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 1-27.

On étudie les groupes discrets $\Lambda$ dont les duaux se plongent dans un groupe quantique compact donné, $\widehat{\Lambda }\subset G$. Dans le cas matriciel $G\subset U_n^{+}$ la condition de plongement est équivalente à l’existence d’une application quotient ${\Gamma }_U\to \Lambda$, où $F=\{{\Gamma }_U\mid U\in U_n\}$ est une certaine famille de groupes associés à $G$. On dévéloppe ici un nombre de techniques pour le calcul de $F$, en partie inspirées pas la classification de Bichon des sous-groupes $\widehat{\Lambda }\subset S_n^{+}$. Ces résultats sont motivés pas la notion de groupe quantique d’isométrie de Goswami, car une variété Riemannienne compacte et connexe ne peut pas avoir des isométries quantiques venant du dual d’un groupe non-abélien.

We study the discrete groups $\Lambda$ whose duals embed into a given compact quantum group, $\widehat{\Lambda }\subset G$. In the matrix case $G\subset U_n^{+}$ the embedding condition is equivalent to having a quotient map ${\Gamma }_U\to \Lambda$, where $F=\{{\Gamma }_U\mid U\in U_n\}$ 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 $\widehat{\Lambda }\subset S_n^{+}$. 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.

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