Quantum isometry groups of noncommutative manifolds associated to group C*-algebras

TL;DR

This paper computes quantum isometry groups of noncommutative manifolds derived from group C*-algebras, revealing new quantum groups and extending understanding of symmetries in noncommutative geometry.

Contribution

It introduces the quantum isometry group for the standard spectral triple on group C*-algebras, including explicit calculations for free groups, and explores related quantum symmetries.

Findings

Quantum isometry group of C*(G) identified as a new quantum group extension.

Explicit computation of quantum isometry group for free groups on n-generators.

Construction of Laplacian and analysis of orientation-preserving quantum isometries.

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.