# Quantum Group of Isometries in Classical and Noncommutative Geometry

## TL;DR

This paper introduces a quantum generalization of the classical isometry group for Riemannian manifolds, establishing the existence of a universal quantum isometry group for both classical and noncommutative geometries, with explicit examples.

## Contribution

It defines a notion of smooth, isometric actions by compact quantum groups and proves the existence of a universal quantum isometry group, extending classical symmetry concepts to noncommutative spaces.

## Key findings

- Existence of a universal quantum isometry group for spectral triples.
- Construction of a spectral triple on forms equivariant under the quantum isometry group.
- Explicit descriptions of quantum isometry groups for tori, including the quantum double torus.

## Abstract

We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. In fact, we identify the quantum isometry group with the universal object in a bigger category, namely the category of `quantum families of smooth isometries', defined along the line of Woronowicz and Soltan. We also construct a spectral triple on the Hilbert space of forms on a noncommutative manifold which is equivariant with respect to a natural unitary representation of the quantum isometry group. We give explicit description of quantum isometry groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in \cite{hajac} as the universal quantum group of holomorphic isometries of the noncommutative torus.