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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.
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Connes, A.: Cyclic cohomology, quantum group symmetries and the local index formula for SU q ( 2 ) {\rm SU}_{q}(2) , J. Inst. Math. Jussieu 3
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Van Daele, A.: Notes on Compact Quantum Groups, arXiv:math/9803122
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