Quantum Isometry Groups: Examples and Computations

Bhowmick, Jyotishman; Goswami, Debashish

doi:10.1007/s00220-008-0611-5

Cited by 50 publications

(98 citation statements)

References 14 publications

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“…However, we give only some computational details for the first example, and for the rest, the reader is referred to a companion article ( [3]). …”

Section: Examples and Computationsmentioning

confidence: 99%

“…The proof of existence and uniqueness of such a universal quantum group is more or less identical to the proof of existence and uniqueness of QISO. We call G the quantum group of "holomorphic" isometries, and observe in the theorem stated below without proof (see [3]) that this quantum group is nothing but the quantum double torus studied in [11]. Then (B, ∆ 0 ) is a compact quantum group and it has an action α 0 on A θ given by…”

Section: Theorem 31mentioning

confidence: 99%

“…The (co)-action on the generators U, V (say) of A θ are given by the following : From the co-associativity condition, the co-product of QISO(A θ ) can easily be calculated. For the detailed description of the coproduct, counit, antipode and study of the representation theory of QISO(A θ ), the reader is referred to [3]. It is interesting to mention here that the quantum isometry group of A θ is a Rieffel type deformation of the isometry group (which is same as the quantum isometry group) of the commutative two-torus.…”

Section: Theorem 31mentioning

confidence: 99%

“…In a companion article [3] with J. Bhowmick, we provide explicit computations of quantum isometry groups of a few classical and noncommutative manifolds. However, we briefly quote some of main results of [3] in the present article.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Group of Isometries in Classical and Noncommutative Geometry

Goswami

2008

Commun. Math. Phys.

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201

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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. 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 [11] as the universal quantum group of holomorphic isometries of the noncommutative torus.

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“…However, we give only some computational details for the first example, and for the rest, the reader is referred to a companion article ( [3]). …”

Section: Examples and Computationsmentioning

confidence: 99%

Section: Theorem 31mentioning

confidence: 99%

Section: Theorem 31mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum Group of Isometries in Classical and Noncommutative Geometry

Goswami

2008

Commun. Math. Phys.

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201

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“…In [5] D. Goswami introduced the notion of a quantum isometry group of a noncommutative manifold. A number of examples of these quantum groups were found in [5] and [2]. The famous quantum spheres of Podleś ([8]) have been given a structure of a noncommutative manifold in [4], and it was proved by J. Bhowmick and D. Goswami in [1] that the quantum isometry groups of these noncommutative manifolds are precisely the quantum SO.3/ groups.…”

Section: Introductionmentioning

confidence: 99%