Non-compact quantum groups associated with Abelian subgroups
type: Item
instance of: scholarly article
title: Non-compact quantum groups associated with Abelian subgroups
zbMATH Open document ID: 0857.17014
DOI: 10.1007/BF02103775
author: Marc A. Rieffel
published in: Communications in Mathematical Physics
publication date: 1997-03-11T00:00:00Z
review text: Soit \(G\) un groupe de Lie d'algèbre de Lie \({\mathfrak g}\) et soit \({\mathfrak h}\) une sous-algèbre de Lie abélienne de \({\mathfrak g}\). Au début de cet article l'auteur rappelle comment on peut associer à tout \(r\) dans \({\mathfrak h}\wedge{\mathfrak h}\) une structure de Poisson compatible sur \(G\). Par ailleurs dans [Deformation quantization for actions of \(\mathbb{R}^d\), Mem. Am. Math. Soc. 506 (1993; Zbl 0798.46053)] l'auteur a développé une méthode de construction d'une déformation d'une \(C^*\)-algèbre dans la direction de n'importe quel crochet de Poisson provenant d'une action de \(\mathbb{R}^d\) \((d\geq 2)\). Il utilise alors cette méthode pour construire le groupe quantique associé à la structure de Poisson correspondante à \(r\in{\mathfrak h}\wedge{\mathfrak h}\). Cette quantification sera préférée, c'est-à-dire, seulement le produit des fonctions sur \(G\) sera déformé. Cette dernière construction a été détaillée dans [Contemp. Math. 145, 465-491 (1993; Zbl 0795.17017)] dans le cas où \(G\) est compact. Ce travail est intéressant car il fournit plusieurs nouveaux exemples de groupes quantiques.
reviewed by: M. Masmoudi
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 46L87
zbMATH DE Number: 850561
zbMATH Keywords: deformation of \(C^*\)-algebra
zbMATH Keywords: quantum groups
zbMATH Keywords: quantum deformations of compact groups
zbMATH Keywords: Poisson structure
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 850561