Quantum Group of Isometries in Classical and Noncommutative Geometry
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In this paper, a quantum generalization of the notion of the group of Riemannian isometries for a compact manifold is introduced, by introducing a natural notion of smooth and isometric action by a compact quantum group on a non-commutative manifold described by spectral triples.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.read more
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Liberation of orthogonal Lie groups
Teodor Banica,Roland Speicher +1 more
TL;DR: In this paper, it was shown that under suitable assumptions, there is a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case.
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Quantum Isometry Groups: Examples and Computations
TL;DR: In this article, the quantum isometry group of a non-commutative manifold has been explicitly computed for a number of classical as well as non-commodity manifolds including the spheres and the tori.
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Quantum Isometries and Noncommutative Spheres
Teodor Banica,Debashish Goswami +1 more
TL;DR: In this article, the authors introduce the half-liberated sphere and the free sphere, which have the property that the corresponding quantum isometry group is easy, in the representation theory sense.
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The full classification of orthogonal easy quantum groups
Sven Raum,Moritz Weber +1 more
TL;DR: In this paper, Banica and Speicher gave a definition of compact matrix quantum groups generalizing compact Lie groups in the setting of noncommutative geometry, which they called easy quantum groups.
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TL;DR: In this article, the authors construct and study compact quantum groups from free products of C======*-algebras, and discover two mysterious classes of natural compact groups, A====== u¯¯ �(m) and A====== o¯¯ ��(m), which are non-isomorphic to each other for different m's, and are not obtainable by the ordinary quantization method.
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TL;DR: In this paper, the quantum automorphism groups of finite spaces were determined, i.e., compact matrix quantum groups in the sense of Woronowicz, and the quantum groups were defined.
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Notes on Compact Quantum Groups
Ann Maes,Alfons Van Daele +1 more
TL;DR: In this paper, the authors follow the approach of Woronowicz and treat the compact quantum groups in the C ∗ -algebra framework and develop the theory of locally compact groups.