Quantum isometry groups of noncommutative manifolds associated to groupC∗-algebras

2010/02/28 by Jyotishman Bhowmick, Adam Skalski · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Compact quantum group #Geometry #Group (periodic table) #Isometry (Riemannian geometry) #Isometry group #Mathematics #Noncommutative geometry #Orientation (vector space) #Physics #Pure mathematics #Quantum #Quantum group #Quantum mechanics #math.OA #math.QA #msc:16W30 #msc:46L89 #msc:58B32

paper · pdf · doi:10.1016/j.geomphys.2010.05.007

23 pages, v2 corrects a few misprints and adds more explanatory remarks. The paper will appear in the Journal of Geometry and Physics

arxiv created 2010/05/10 · openalex publication_date 2010/06/12 · arxiv updated 2015/05/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

Let G be a finitely generated discrete group. The standard spectral triple on the group C*-algebra C*(G) is shown to admit the quantum group of orientation preserving isometries. This leads to new examples of compact quantum groups. In particular the quantum isometry group of the C*-algebra of the free group on n-generators is computed and turns out to be a quantum group extension of the quantum permutation group A_2n of Wang. The quantum groups of orientation and real structure preserving isometries are also considered and construction of the Laplacian for the standard spectral triple on C*(G) discussed.

Citations

• Spectral triples for AF C*-algebras and metrics on the Cantor set
• Notes on Compact Quantum Groups
• Quantum Group of Isometries in Classical and Noncommutative Geometry
• Quantum Isometry Group for Spectral Triples with Real Structure
• Quantum Isometry Groups of 0- Dimensional Manifolds

Cited by

• Lévy Processes on Quantum Permutation Groups