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. The idea of ‘quantum families’ (due to Woronowicz and Soltan) are relevant to our construction. A number of explicit examples are given and possible applications of our results to the problem of constructing quantum group equivariant spectral triples are discussed.

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Correspondence to Debashish Goswami.

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Supported in part by the Indian National Academy of Sciences.

Communicated by A. Connes

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Goswami, D. Quantum Group of Isometries in Classical and Noncommutative Geometry. Commun. Math. Phys. 285, 141 (2009). https://doi.org/10.1007/s00220-008-0461-1

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Keywords

  • Dirac Operator
  • Quantum Group
  • Noncommutative Geometry
  • Compact Quantum Group
  • Spectral Triple