Quantum Group of Isometries in Classical and Noncommutative Geometry

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.

improvements: (i) more general approach through quantum families of isometries, following Woronowicz and Soltan's ideas; (ii) construction of equivaraint spectral triples. also some minor changes. Improved and enlarged version of ICTP preprint IC/2007/017

References in corpus (4)

• Quantum Symmetry Groups of Finite Spaces
• The Dirac operator on SU_q(2)
• Supersymmetric quantum theory and non-commutative geometry
• Quantum Double-Torus

Cited by in corpus (17)

• Quantum isometries and noncommutative spheres
• Faithful compact quantum group actions on connected compact metrizable spaces
• Quantum isometry groups of noncommutative manifolds associated to group C*-algebras
• Quantum Isometries of the finite noncommutative geometry of the Standard Model
• Liberations and twists of real and complex spheres
• Levi-Civita connections for a class of spectral triples
• Quantum isometry groups of symmetric groups
• Some counterexamples in the theory of quantum isometry groups
• Quantum symmetries of the twisted tensor products of C*-algebras
• Quantum Brownian Motion on noncommutative manifolds: construction, deformation and exit times
• Faithful actions of locally compact quantum groups on classical spaces
• Rigidity on Quantum Symmetry for a Certain Class of Graph C*-algebras
• Deformations of spectral triples and their quantum isometry groups via monoidal equivalences
• Existence and rigidity of quantum isometry groups for compact metric spaces
• Quantum isomorphism of $2$-graphs
• Quantum symmetries on noncommutative complex spheres with partial commutation relations
• Quantum symmetry groups of noncommutative tori