Quantum group of isometries in classical and noncommutative geometry

Goswami, Debashish (2009) Quantum group of isometries in classical and noncommutative geometry Communications in Mathematical Physics, 285 . Article ID 141. ISSN 0010-3616

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Official URL: http://link.springer.com/article/10.1007/s00220-00...

Related URL: http://dx.doi.org/10.1007/s00220-008-0461-1


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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