Quantum isometries and group dual subgroups

Teodor Banica; Jyotishman Bhowmick; Kenny De Commer

doi:10.5802/ambp.303

Quantum isometries and group dual subgroups
[Isométries quantiques et duaux de groupes]

Teodor Banica ¹ ; Jyotishman Bhowmick ² ; Kenny De Commer ¹

¹ Department of Mathematics Cergy-Pontoise University 95000 Cergy-Pontoise FRANCE
² Department of Mathematics University of Oslo Blindern, N-0315 Oslo NORWAY

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

Résumé
Abstract

On étudie les groupes discrets $Λ$ dont les duaux se plongent dans un groupe quantique compact donné, $\hat{Λ} \subset G$ . Dans le cas matriciel $G \subset U_{n}^{+}$ la condition de plongement est équivalente à l’existence d’une application quotient $Γ_{U} \to Λ$ , où $F = {Γ_{U} ∣ U \in U_{n}}$ 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 $\hat{Λ} \subset S_{n}^{+}$ . 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 $Λ$ whose duals embed into a given compact quantum group, $\hat{Λ} \subset G$ . In the matrix case $G \subset U_{n}^{+}$ the embedding condition is equivalent to having a quotient map $Γ_{U} \to Λ$ , where $F = {Γ_{U} ∣ U \in U_{n}}$ 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 $\hat{Λ} \subset S_{n}^{+}$ . 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.

MR Zbl

DOI : 10.5802/ambp.303

Classification : 58J42
Mots clés : Quantum isometry, Diagonal subgroup

Affiliations des auteurs :

Teodor Banica ¹ ; Jyotishman Bhowmick ² ; Kenny De Commer ¹

¹ Department of Mathematics Cergy-Pontoise University 95000 Cergy-Pontoise FRANCE
² Department of Mathematics University of Oslo Blindern, N-0315 Oslo NORWAY

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Teodor Banica; Jyotishman Bhowmick; Kenny De Commer. Quantum isometries and group dual subgroups. Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 1-27. doi : 10.5802/ambp.303. https://ambp.centre-mersenne.org/articles/10.5802/ambp.303/

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